The length of closed geodesics on random Riemann Surfaces

a r X i v :0708.2841v 1 [m a t h -p h ] 21 A u g 2007Geodesics and Geodesic Deviation in aStringy Charged Black HoleRagab M.Gad 1Mathematics Department,Faculty of Science,Minia University,61915El-Minia,EGYPT.AbstractThe radial motion along null geodesics in the charged black hole space-times,in particular,the Reissner-Nordstr¨o m and stringy charged black holes are studied.We analyze the properties of the effective potential.The circular photon orbits in these space-times are investi-gated.We find that the radius of circular photon orbits in both charged black holes are different and differ from that given in Schwarzschild space-time.We Study the physical effects of the gravitational field between two test particles in stringy charged black hole and compare the results with that given in Schwarzschild and Reissner-Nordstr¨o m black holes.1IntroductionThe well known static,spherically symmetric black hole solutions in vacuum of Einstein’s general relativity are given by the charged Reissner-Nordstr¨o m and uncharged Schwarzschild solutions.In the non-vacuum case of Einstein’s general relativity,several black hole solutions are known [1]-[3].One of them is the stringy charged black hole discovered by Garfinkle,Horowitz and Strominger [3].The study of timelike and null geodesics,the paths of freely moving particles and photons,is the key to understanding the physical importance of a given space-time.We wish to investigate in this paper the properties of the stringy charged black hole by studying its geodesic structure,that is,from the motion of photons.We compare the results by the aforementioned solutions in vacuum.The space-time under consideration is almost identical to Schwarzschild space-time.The only differences are1.The gravitational energy,using Møller’s prescription [4],depends on the mass parameter M and on the charge Q (see Gad [5]),while inthe Schwarzschild black hole is given only by the mass parameter M(see Xulu[6]).2.The areas of the spheres of constant r and t depend on the charge Q. Thefirst aim of this paper is to sustain the second deference by analyzing the effective potential of radial motion along null geodesics.Compared to the Reissner-Nordstr¨o m black hole,the stringy charged black hole exhibits several different properties[7]-[10].For example,first, this solution has only one horizon at r=2M and not two as is the case for the Reissner-Nordstr¨o m2.Second,this solution has singularities at r=α2The Reissner-Nordstr¨o m has two horizons given by the quadratic equation r2−2Mr+e2=0.2[20]and[7]).There are two metrics in this theory,which are called the sigma-model(or string)metric and Einstein metric.For uncharged static, spherically symmetric black hole,the solution in the low energy is the same as the Schwarzschild solution.This is,however,not the case when the black hole is charged.We are interested to investigate the geodesic curves of a static spherically symmetric charged black hole.The line element representing this space-time is given by Grafinkle et al[3].ds2=−(1−2Mr)−1dr2+(1−αM,M and Q are,respectively,mass and charged parameters;Φ0is the asymp-totic value of dilatonfield.The equations and constraint for geodesics are given as¨x a+Γa bc˙x b˙x c=0,g ab˙x a˙x b=ε,where a superposed dot stands for a derivative with respect to the affine parameterτassociated to the geodesic,x a are the coordinates of a space-time point on the geodesic andε=−1or0,for timelike or null geodesics, respectively.The geodesic equations for the line element(2.1)are given by¨t+2Mr(r−2M)˙r2+(r−2M)(α−2r)2rsin2θ˙φ2+ M(r−2M)r(α−r)˙r˙θ−sinθcosθ˙φ2=0,(2.4)¨φ+(α−2r)The constraint of timelike or null geodesics for the line element(2.1)is given by−(1−2M r)−1˙r2+(1−α2,φ(τ0))and for allτ:θ=π(r−2M)),(2.7)˙φ=c2dτ(τ0)for someτ0∈ℜ.In the case of Schwarzschild andReissner-Nordstr¨o m space-times,Clarke[18]and also Wald[19]demonstrate that c1represents the total energy,E,per unit rest mass of a particle as measured by a static observer,and c2represents the angular momentum, L,per unit mass of a particle(see also[11]).We recognize the constant εto represent the rest energy per unit mass for massive particles(timelike curve,ε=−1)or the rest energy for massless particles(null curves,ε=0), travelling along the given geodesic[11,12].Using equations(2.7),(2.8)andθ=πr )[−ε+L2r )[−ε+L2Equation (2.10)is in the form of the equation of a one-dimensional problem for a particle in a potential field V (r ).Since the left side of equation (2.9)is positive or zero,the energy E of the trajectory must not be less than the potential V .So for an orbit of a given E ,the radial range is restricted to those radii for which V is smaller than E .In the following we will investigate the circular photon orbits in the stringy charged black hole by analyzing the properties of the effective po-tential.In the case of the photon trajectories,putting ε=0,the effective potential (2.11)takes the formV 2(r )=(1−2Mr (r −α)].(2.12)Differentiating equation (2.10)with respect to τgives2(drdτ2)=−dV 2(r )dτ,ord 2r2ddr(1−2Mr (r −α)],and getr =1(6M −α)2−16Mα.(2.14)For α=(14−4√10)M ,this implies negative root,which hasno physical significance).For α=6M and α>6M no circular is possible.For α<6M the larger of two roots given by equation (2.14)locates the minimum of the potential-energy curve V 2(r )defined by equation (2.12),while the smaller root locates the maximum of the potential-energy curve.Therefore,the circular orbit of the larger radius will be stable in the contrast to circular orbit of the smaller radius which will be unstable.We notice that,when α=0,that is the Schwarzschild case,the ra-dius is r =3M which is the same radius obtained by Schutz [13]in the Schwarzschild black hole.53Reissner-Nordstr¨o m metricA well-known simple solution of the Einstein-Maxwell equation is the Reissner-Nordstr¨o m solution.This solution represents a non-rotating charged black hole.Discussions of the basic properties of this solution can be found in many places including works by Chandrasekhar[14],and Hawking and Ellis [15].The metric is defined on a four-dimensional manifold and its typically written in the formds2=−(1−2mr2)dt2+(1−2mr2)−1dr2+r2(dθ2+sin2θdφ2),(3.1)where m represents the gravitational mass and e the electric charge of the body.The equation of geodesics and the constraint of geodesic for the line element(3.1)are given as¨t+2(−m+e2r2−2mr+e2˙r˙t=0,(3.2)¨θ+2r˙r˙φ+2cotθ˙θ˙φ=0,(3.4)−(1−2m r2)˙t2+(1−2m r2)−1˙r2+r2(˙θ2+sin2θ˙φ2)=ε.(3.5) We will assume,as section2,that the orbit is in theθ=π2and˙θ=0initially,then¨θ=0and the orbitremains in this plane.Equations(3.2)and(3.4)can be integrated directly,giving˙t=c3r2r2,(3.7)where the integrating constant c3represents the energy,¯E,(at r→∞)of a test particle and c4the angular momentum,¯L.Substituting(3.6)and(3.7)in equation(3.5)and using the condition ˙θ=0,we get˙r2=¯E2−(1−2m r2)[−ε+¯L2Equation (3.8)can be written as˙r 2=¯E2−¯V 2(r ),(3.9)where ¯V(r )is the ”effective potential”defined by ¯V 2(r )=(1−2mr 2)[−ε+¯L2r+e 2r 2].(3.11)Differentiating (3.9)with respect to τ,as in the previous section,we getd 2r2ddr(1−2mr 2)[¯L221±9m 2.(3.13)This equation shows that the two radii are identical for e22)and don’t exist at all for e2m .Fore 24Geodesic DeviationIn this section,we use the tidal forces between free test particles falling in a gravitationalfield to investigate the different properties between the stringy charged and vacuum black holes.Consider a sphere of two non-interacting particles falling freely towards the center of the Earth.Each particle moves on a straight line,but nearer the Earth fall faster because the gravitational attraction is stronger.This means that the sphere does not remain a sphere but is distorted into an ellipsoid with the same volume.The same effect occurs in a body falling towards a spherical object in general relativity,but if the object is a black hole the effect becomes infinite as the singularity is reached.Jacobi vector fields provide the connection between the behavior of nearby particles and curvature,via the equation of geodesic deviation(Jacobi equation)D2ηaDτe aα=0,α=1,2,3),e a0will remain equal v a,and e a1,e a2,e a3will remain to orthogonal to v a(see[16]p.80).The frame e a0,e a1,e a2,e a3 is called”parallel transported”(PT)frame.The orthogonal connecting vector,ηa,between two neighboring timelike geodesics may be expressed as ηa=ηαe aα(η0=e0αηα=0).The Jacobi vectorfieldsηa satisfy the following equationDηαDτ2+˜R a bdc eαa v b v c e dβηβ=0,(4.3) whereηαare the space-like components of the orthogonal connecting vector ηa connecting two neighboring particles in free fall;η0=0.The tilde8denotes components in the PT frame and the components of the Riemann tensor˜R a bdc are given by˜R abdc=e a e e f b e g c e h d R e fgh.(4.4) From(2.1)the frame e a b in Reissner-Nordstr¨o m metric is given by:e a0=(1−2m2(0,0,0,1),e a1=(1−2m2(1,0,0,0),e a2=(1−α2r )−1r sinθ(0,0,1,0).(4.5)The components ofηαcan be written as followsηα=(η1,η2,η3)=(ηr,ηθ,ηφ).Using(4.4),(4.5),v a=e a0and the components of Riemann tensor for the metric(2.1)(see appendix),in(4.3),we getD2ηrr3ηr,D2ηθ2r3(r−α)ηθ,D2ηφ2r3(r−α)ηφ,(4.6)In order to write equation(4.6)in terms of ordinary derivative,we must evaluate the second covariant derivative D2Dτ=dηαdτ2=2Mdτ2=M(α−2r)d2ηφηφ.(4.10)2r3(r−α)Equation(4.8)indicates tidal force in radial direction will stretch an ob-server falling in thisfluid.To keep the line element(2.1)to be in Lorentzian metric,αmust be less than r.Therefore equations(4.9)and(4.10)indicate a pressure or compression in the transverse directions.ConclusionIn this paper we have studied the circular photon orbits in charged black holes by analyzing the properties of effective potential.Considering the light-like geodesics,we classified and analyzed the different cases between the stringy charged black hole and the vacuum solutions.These differences arised by considering the orbits associated with stable and unstable circular orbits.In the context of the Schwarzschild geometry there is an unstable circular orbit which is always at the same radius,r=3M.In the Reissner-Nordstr¨o m and the stringy charged black holes,there are two radii,the circular orbit of the larger radius will be stable while that of the smaller radius will be unstable.Equations(4.8)-(4.10)provide the explicit expressions of the relative ac-celerations in a stringy charged black hole.Two comments are worth making about expression(4.8).First there is no divergence in the radial direction at r=2m.Secondly,the tidalfield at the horizon in radial direction is larger for smaller black hole.This is simply becaused2ηrηr∼1r3on the quantity e2−2mr to indicate compression or tension in these direc-tions.AppendixWe use(x0,x1,x2,x3)=(t,r,θ,φ)so that the non-vanishing Christoffel symbols of the second kind of the line element(2.1))areΓ111=M2r(α−r),Γ122=(r−2M)(α−2r2r(α−r,Γ133=(r−2M)(α−2rr3,Γ323=cotθ,Γ010=−M4r2(α−r), R1313=2M(r−α)(2r−α)+α2(2M−r)r4,R2323=8Mr(r−α)−α2(r−2M)2r4(r−α).(4.2)References[1]J.D.Bekenstein,Ann.Phys.(N.Y.),91,75(1975).[2]E.Ayon-Beato and A.Garcia,Gem.Relat.Grav.31,629(1999).[3]D.Garfinkle,G.T.Horowitz and A.Strominger,Phys.Rev.D43,3140(1991);D45,3888(E)(1992).[4]C.Møller,Ann.Phys.(N.Y.),4,347(1958).[5]R.M.Gad,Astrophys.Space Sci.295,459(2004).[6]S.S.Xulu,Astrophys.Space Sci.283,23(2003).[7]J.Preskill,P.Schwarz,A.Shapere,S.Trivedi and F.Wilczek,Mod.Phys.Lett.,6,2353(1991).11[8]J.A.Harvey and A.Strominger,”Quantum aspects of black holes”,Preprint EFI-92-41,hep-th/9209055.[9]F.A.E.Piran,Phys.Rev.105,1089(1957).[10]K.P.Tod,Proc.R.Soc.Lond.A388,467(1983).[11]C.Misner,K.Thorne and J.Wheeler,(1973),”Gravitation”,Freeman,San Francisco.[12]R.Adler,M.Bazin and M.Schiffer,(1975),”Introduction to GeneralRelativity”,(McGrow-Hill,New York,2nd.ed.).[13]B.F.Schutz,(1985),”A First Course in General Relativity”(Cam-bridge Uni.Press,Cambridge,London,New York,New Rochelle,Mel-bourne Sydney).[14]S.Chandrasekhar,(1983),”The Mathematical Theory of Black Holes”,(Oxford Uni.Press,Cambridge,England).[15]S.W.Hawking and G.F.R.Ellis,”The Larger Scale Structure ofSpace-time”,(Cambridge Uni.Press,Cambridige).[16]R.D’Invermo,”Introducing Einstein’s Relativity”,Oxford UniversityPress,New York,(1992).[17]M.Abdel-Megied and R.M.Gad,Chaos,Solitons and Fractals,23,313(2005).[18]C.J.S.Clark,(1979),”Elementary General relativity”,(EdwaedArnold,London).[19]R.M.Wald,(1984),”General Relativity”,(Chicago and London).[20]A.Shapere,S.Trivedi and F.Wilczek,Mod.Phys.Lett.A6,2677(1991).12。

a rX iv:mat h /47161v3[mat h.GT]22O ct25SHRINKWRAPPING AND THE TAMING OF HYPERBOLIC 3–MANIFOLDS DANNY CALEGARI AND DAVID GABAI 0.I NTRODUCTION During the period 1960–1980,Ahlfors,Bers,Kra,Marden,Maskit,Sullivan,Thurston and many others developed the theory of geometrically finite ends of hy-perbolic 3–manifolds.It remained to understand those ends which are not geo-metrically finite;such ends are called geometrically infinite .Around 1978William Thurston gave a conjectural description of geometrically infinite ends of complete hyperbolic 3–manifolds.An example of a geometrically infinite end is given by an infinite cyclic covering space of a closed hyperbolic 3-manifold which fibers over the circle.Such an end has cross sections of uniformly bounded area.By contrast,the area of sections of geometrically finite ends grow exponentially in the distance from the convex core.For the sake of clarity we will assume throughout this introduction that N =H 3/Γwhere Γis parabolic free.Precise statements of the parabolic case will be given in §7.Thurston’s idea was formalized by Bonahon [Bo]and Canary [Ca]with the fol-lowing.Definition 0.1.An end E of a hyperbolic 3-manifold N is simply degenerate if it has a closed neighborhood of the form S ×[0,∞)where S is a closed surface,and there exists a sequence {S i }of CAT (−1)surfaces exiting E which are homotopic to S ×0in E .This means that there exists a sequence of maps f i :S →N such that the induced path metrics induce CAT (−1)structures on the S i ’s,f (S i )⊂S ×[i,∞)and f is homotopic to a homeomorphism onto S ×0via a homotopy supported in S ×[0,∞).Here by CAT (−1),we mean as usual a geodesic metric space for which geo-desic triangles are “thinner”than comparison triangles in hyperbolic space.If themetrics pulled back by the f i are smooth,this is equivalent to the condition that the Riemannian curvature is bounded above by −1.See [BH]for a reference.Note that by Gauss–Bonnet,the area of a CAT (−1)surface can be estimated from its Euler characteristic;it follows that a simply degenerate end has cross sections of uniformly bounded area,just like the end of a cyclic cover of a manifold fibering over the circle.Francis Bonahon [Bo]observed that geometrically infinite ends are exactly those ends possessing an exiting sequence of closed geodesics.This will be our working definition of such ends throughout this paper.2DANNY CALEGARI AND DAVID GABAIThe following is our main result.Theorem0.2.An end E of a complete hyperbolic3-manifold N withfinitely generated fundamental group is simply degenerate if there exists a sequence of closed geodesics exit-ing E.Consequently we have,Theorem0.3.Let N be a complete hyperbolic3-manifold withfinitely generated funda-mental group.Then every end of N is geometrically tame,i.e.it is either geometrically finite or simply degenerate.In1974Marden[Ma]showed that a geometricallyfinite hyperbolic3-manifold is topologically tame,i.e.is the interior of a compact3-manifold.He asked whether all complete hyperbolic3-manifolds withfinitely generated fundamental group are topologically tame.This question is now known as the Tame Ends Conjecture or Marden Conjecture.Theorem0.4.If N is a complete hyperbolic3-manifold withfinitely generated funda-mental group,then N is topologically tame.Ian Agol[Ag]has independently proven Theorem0.4.There have been many important steps towards Theorem0.2.The seminal re-sult was obtained by Thurston([T],Theorem9.2)who proved Theorems0.3and 0.4for certain algebraic limits of quasi Fuchsian groups.Bonahon[Bo]estab-lished Theorems0.2and0.4whenπ1(N)is freely indecomposible and Canary [Ca]proved that topological tameness implies geometrical tameness.Results in the direction of0.4were also obtained by Canary-Minsky[CaM],Kleineidam–Souto[KS],Evans[Ev],Brock–Bromberg–Evans–Souto[BBES],Ohshika,Brock–Souto[BS]and Souto[So].Thurstonfirst discovered how to obtain analytic conclusions from the existence of exiting sequences of CAT(−1)surfaces.Thurston’s work as generalized by Bonahon[Bo]and Canary[Ca]combined with Theorem0.2yields a positive proof of the Ahlfors’Measure Conjecture[A2].Theorem0.5.IfΓis afinitely generated Kleinian group,then the limit set LΓis either S2∞or has Lebesgue measure zero.If LΓ=S2∞,thenΓacts ergodically on S2∞.Theorem0.5is one of the many analytical consequences of our main result. Indeed Theorem0.2implies that a complete hyperbolic3-manifold N withfinitely generated fundamental group is analytically tame as defined by Canary[Ca].It follows from Canary that the various results of§9[Ca]hold for N.Our main result is the last step needed to prove the following monumental result,the other parts being established by Alhfors,Bers,Kra,Marden,Maskit, Mostow,Prasad,Sullivan,Thurston,Minsky,Masur–Minsky,Brock–Canary–Minsky, Ohshika,Kleineidam–Souto,Lecuire,Kim–Lecuire–Ohshika,Hossein–Souto and Rees.See[Mi]and[BCM].Theorem0.6(Classification Theorem).If N is a complete hyperbolic3-manifold with finitely generated fundamental group,then N is determined up to isometry by its topolog-ical type,the conformal boundary of its geometricallyfinite ends and the ending lamina-tions of its geometrically infinite ends.SHRINKWRAPPING AND THE TAMING OF HYPERBOLIC3–MANIFOLDS3 The following result was conjectured by Bers,Sullivan and Thurston.Theorem0.4is one of many results,many of them recent,needed to build a proof.Major contributions were made by Alhfors,Bers,Kra,Marden,Maskit,Mostow,Prasad, Sullivan,Thurston,Minsky,Masur–Minsky,Brock–Canary–Minsky,Ohshika,Kleineidam–Souto,Lecuire,Kim–Lecuire–Ohshika,Hossein–Souto,Rees,Bromberg and Brock–Bromberg.Theorem0.7(Density Theorem).If N=H3/Γis a completefinitely generated3-manifold withfinitely generated fundamental group,thenΓis the algebraic limit of geo-metricallyfinite Kleinian groups.The main technical innovation of this paper is a new technique called shrinkwrap-ping for producing CAT(−1)surfaces in hyperbolic3-manifolds.Historically,such surfaces have been immensely important in the study of hyperbolic3-manifolds,e.g.see[T],[Bo],[Ca]and[CaM].Given a locallyfinite set∆of pairwise disjoint simple closed curves in the3-manifold N,we say that the embedded surface S⊂N is2-incompressible rel.∆ifevery compressing disc for S meets∆at least twice.Here is a sample theorem. Theorem0.8(Existence of shrinkwrapped surface).Let M be a complete,orientable, parabolic free hyperbolic3–manifold,and letΓbe afinite collection of pairwise disjoint sim-ple closed geodesics in M.Further,let S⊂M\Γbe a closed embedded2–incompressible surface rel.Γwhich is either nonseparating in M or separates some component ofΓfrom another.Then S is homotopic to a CAT(−1)surface T via a homotopyF:S×[0,1]→Msuch that(1)F(S×0)=S(2)F(S×t)=S t is an embedding disjoint fromΓfor0≤t<1(3)F(S×1)=T(4)If T′is any other surface with these properties,then area(T)≤area(T′)We say that T is obtained from S by shrinkwrapping rel.Γ,or ifΓis understood,T is obtained from S by shrinkwrapping.In fact,we prove the stronger result that T isΓ–minimal(to be defined in§1)which implies in particular that it is intrinsically CAT(−1)Here is the main technical result of this paper.Theorem0.9.Let E be an end of the complete orientable hyperbolic3-manifold N withfinitely generated fundamental group.Let C be a3-dimensional compact core of N,∂E Cthe component of∂C facing E and g=genus(∂E C).If there exists a sequence of closed geodesics exiting E,then there exists a sequence{S i}of CAT(−1)surfaces of genus g exit-ing E such that each{S i}is homologically separating in E.That is,each S i homologically separates∂E C from E.Theorem0.4can now be deduced from Theorem0.9and Souto[So];however,we prove that Theorem0.9implies Theorem0.4using only3-manifold topologyand elementary hyperbolic geometry.The proof of Theorem0.9blends elementary aspects of minimal surface theory, hyperbolic geometry,and3-manifold topology.The method will be demonstrated4DANNY CALEGARI AND DAVID GABAIin§4where we give a proof of Canary’s theorem.Thefirst time reader is urged to begin with that section.This paper is organized as follows.In§1and§2we establish the shrinkwrap-ping technique forfinding CAT(−1)surfaces in hyperbolic3-manifolds.In§3we prove the existence ofǫ-separated simple geodesics exiting the end of parabolic free manifolds.In§4we prove Canary’s theorem.This proof will model the proof of the general case.The general strategy will be outlined at the end of that section. In§5we develop the topological theory of end reductions in3-manifolds.In§6we give the proofs of our main results.In§7we give the necessary embellishments of our methods to state and prove our results in the case of manifolds with parabolic cusps.Notation0.10.If X⊂Y,then N(X)denotes a regular neighborhood of X in Y and int(X)denotes the interior of X.If X is a topological space,then|X|denotes the number of components of X.If A,B are topological subspaces of a third space, then A\B denotes the intersection of A with the complement of B. Acknowledgements0.11.Thefirst author is grateful to Nick Makarov for some useful analytic discussions.The second author is grateful to Michael Freedman for many long conversations in Fall1996which introduced him to the Tame Ends con-jecture.He thanks Francis Bonahon,Yair Minsky and Jeff Brock for their interest and helpful comments.Part of this research was carried out while he was visiting Nara Women’s University,the Technion and the Institute for Advanced Study.He thanks them for their hospitality.We thank the referees for their many thoughtful suggestions and comments.1.S HRINKWRAPPINGIn this section,we introduce a new technical tool forfinding CAT(−1)surfaces in hyperbolic3–manifolds,called shrinkwrapping.Roughly speaking,given a col-lection of simple closed geodesicsΓin a hyperbolic3–manifold M and an embed-ded surface S⊂M\Γ,a surface T⊂M is obtained from S by shrinkwrapping S rel.Γif it homotopic to S,can be approximated by an isotopy from S supported in M\Γ,and is least area subject to these constraints.Given mild topological conditions on M,Γ,S(namely2–incompressibility,to be defined below)the shrinkwrapped surface exists,and is CAT(−1)with respect to the path metric induced by the Riemannian metric on M.We use some basic analytical tools throughout this section,including the Gauss–Bonnet formula,the coarea formula,and the Arzela–Ascoli theorem.At a number of points we must invoke results from the literature to establish existence of min-imal surfaces([MSY]),existence of limits with area and curvature control([CiSc]), and regularity of the shrinkwrapped surfaces alongΓ([Ri],[Fre]).General refer-ences are[CM],[Js][Fed]and[B].1.1.Geometry of surfaces.For convenience,we state some elementary but fun-damental lemmas concerning curvature of(smooth)surfaces in Riemannian3-manifolds.We use the following standard terms to refer to different kinds of minimal sur-faces:Definition1.1.A smooth surfaceΣin a Riemannian3-manifold is minimal if it is a critical point for area with respect to all smooth compactly supported variations.SHRINKWRAPPING AND THE TAMING OF HYPERBOLIC3–MANIFOLDS5 It is locally least area(also called stable)if it is a local minimum for area with respect to all smooth,compactly supported variations.A closed,embedded surface is globally least area if it is an absolute minimum for area amongst all smooth surfaces in its isotopy class.Note that we do not require that our minimal or locally least area surfaces are complete.Any subsurface of a globally least area surface is locally least area,and a locally least area surface is minimal.A smooth surface is minimal iff its mean curvature vectorfield vanishes identically.For more details,consult[CM],especially chapter 5.The intrinsic curvature of a minimal surface is controlled by the geometry of the ambient manifold.The following lemma is formula5.6on page100of[CM]. Lemma1.2(Monotonicity of curvature).LetΣbe a minimal surface in a Riemannian manifold M.Let KΣdenote the curvature ofΣ,and K M the sectional curvature of M. Then restricted to the tangent space TΣ,1KΣ=K M−6DANNY CALEGARI AND DAVID GABAIfor smallǫ,whereφ(·,0)=Id|∂Σ,andφ(∂Σ,t)for small t is the boundary inΣof the tubular t neighborhood of∂Σ.Then∂Σκdl=−da1a3in the complete simply–connected Riemannian2–manifold of con-stant sectional curvatureκ,where the edges a i a j and a j satisfylength(a i a j)=length(a j)Given a point x∈a1a2on one of the edges of a1a2a3,there is a corresponding point a1a1xa toSHRINKWRAPPING AND THE TAMING OF HYPERBOLIC3–MANIFOLDS7 Definition1.8(Γ–minimal surfaces).Letκ∈R be given.Let M be a complete Riemannian3–manifold with sectional curvature bounded above byκ,and letΓbe an embedded collection of simple closed geodesics in M.An immersionψ:S→MisΓ–minimal if it is smooth with mean curvature0in M\Γ,and is metrically CAT(κ)with respect to the path metric induced byψfrom the Riemannian metric on M.Notice by Lemma1.2that a smooth surface S with mean curvature0in M is CAT(κ),so a minimal surface(in the usual sense)is an example of aΓ–minimal surface.1.3.Statement of shrinkwrapping theorem.Definition1.9(2–incompressibility).An embedded surface S in a3–manifold M disjoint from a collectionΓof simple closed curves is said to be2–incompressible rel.Γif any essential compressing disk for S must intersectΓin at least two points.If Γis understood,we say S is2–incompressible.Theorem1.10(Existence of shrinkwrapped surface).Let M be a complete,orientable, parabolic free hyperbolic3–manifold,and letΓbe afinite collection of pairwise disjoint sim-ple closed geodesics in M.Further,let S⊂M\Γbe a closed embedded2–incompressible surface rel.Γwhich is either nonseparating in M or separates some component ofΓfrom another.Then S is homotopic to aΓ–minimal surface T via a homotopyF:S×[0,1]→Msuch that(1)F(S×0)=S(2)F(S×t)=S t is an embedding disjoint fromΓfor0≤t<1(3)F(S×1)=T(4)If T′is any other surface with these properties,then area(T)≤area(T′)We say that T is obtained from S by shrinkwrapping rel.Γ,or ifΓis understood,T is obtained from S by shrinkwrapping.The remainder of this section will be taken up with the proof of Theorem1.10. Remark1.11.In fact,for our applications,the property we want to use of our surface T is that we can estimate its diameter(rel.the thin part of M)from its Euler characteristic.This follows from a Gauss–Bonnet estimate and the bounded diameter lemma(Lemma1.15,to be proved below).In fact,our argument will show directly that the surface T satisfies Gauss–Bonnet;the fact that it is CAT(−1) is logically superfluous for the purposes of this paper.1.4.Deforming metrics along geodesics.Definition1.12(δ–separation).LetΓbe a collection of disjoint simple geodesics in a Riemannian manifold M.The collectionΓisδ–separated if any pathα:I→M with endpoints onΓand satisfyinglength(α(I))≤δ8DANNY CALEGARI AND DAVID GABAIis homotopic rel.endpoints intoΓ.The supremum of suchδis called the separation constant ofΓ.The collectionΓis weaklyδ–separated ifdist(γ,γ′)>δwheneverγ,γ′are distinct components ofΓ.The supremum of suchδis called the weak separation constant ofΓ.Definition1.13(Neighborhood and tube neighborhood).Let r>0be given.For a point x∈M,we let N r(x)denote the closed ball of radius r about x,and let N<r(x),∂N r(x)denote respectively the interior and the boundary of N r(x).For a closed geodesicγin M,we let N r(γ)denote the closed tube of radius r aboutγ, and let N<r(γ),∂N r(γ)denote respectively the interior and the boundary of N r(γ). IfΓdenotes a union of geodesicsγi,then we use the shorthand notationN r(Γ)= γi N r(γi)Remark1.14.Topologically,∂N r(x)is a sphere and∂N r(γ)is a torus,for suffi-ciently small r.Similarly,N r(x)is a closed ball,and N r(γ)is a closed solid torus. IfΓisδ–separated,then Nδ/2(Γ)is a union of solid tori.Lemma1.15(Bounded Diameter Lemma).Let M be a complete hyperbolic3–manifold. LetΓbe a disjoint collection ofδ–separated embedded geodesics.Letǫ>0be a Margulis constant for dimension3,and let M≤ǫdenote the subset of M where the injectivity radius is at mostǫ.If S⊂M\Γis a2–incompressibleΓ–minimal surface,then there is a con-stant C=C(χ(S),ǫ,δ)∈R and n=n(χ(S),ǫ,δ)∈Z such that for each component S i of S∩(M\M≤ǫ),we havediam(S i)≤CFurthermore,S can only intersect at most n components of M≤ǫ.Proof.Since S is2–incompressible,any point x∈S either lies in M≤ǫ,or is the center of an embedded m–disk in S,wherem=min(ǫ/2,δ/2)Since S is CAT(−1),Gauss–Bonnet implies that the area of an embedded m–disk in S has area at least2π(cosh(m)−1)>πm2.This implies that if x∈S∩M\M≤ǫthenarea(S∩N m(x))≥πm2The proof now follows by a standard covering argument.A surface S satisfying the conclusion of the Bounded Diameter Lemma is some-times said to have diameter bounded by C modulo M≤ǫ.Remark1.16.Note that ifǫis a Margulis constant,then M≤ǫconsists of Margulis tubes and cusps.Note that the same argument shows that,away from the thin part of M and anǫneighborhood ofΓ,the diameter of S can be bounded by a constant depending only onχ(S)andǫ.The basic idea in the proof of Theorem1.10is to search for a least area repre-sentative of the isotopy class of the surface S,subject to the constraint that the track of this isotopy does not crossΓ.Unfortunately,M\Γis not complete,so the prospects for doing minimal surface theory in this manifold are remote.To rem-edy this,we deform the metric on M in a neighborhood ofΓin such a way thatSHRINKWRAPPING AND THE TAMING OF HYPERBOLIC3–MANIFOLDS9 we can guarantee the existence of a least area surface representative with respect to the deformed metric,and then take a limit of such surfaces under a sequence of smaller and smaller such metric deformations.We describe the deformations of interest below.In fact,for technical reasons which will become apparent in§1.8,the defor-mations described below are not quite adequate for our purposes,and we must consider metrics which are deformed twice—firstly,a mild deformation which satisfies curvature pinching−1≤K≤0,and which is totally Euclidean in a neighborhood ofΓ,and secondly a deformation analogous to the kind described below in Definition1.17,which is supported in this totally Euclidean neighbor-hood.Since the reason for this“double perturbation”will not be apparent until §1.8,we postpone discussion of such deformations until that time.Definition1.17(Deforming metrics).Letδ>0be such thatΓisδ–separated. Choose some small r with r<δ/2.For t∈[0,1)we define a family of Riemannian metrics g t on M in the following manner.The metrics g t agree with the hyperbolic metric away from somefixed tubular neighborhood N r(Γ).Leth:N r(1−t)(Γ)→[0,r(1−t)]be the function whose value at a point p is the hyperbolic distance from p to Γ.We define a metric g t on M which agrees with the hyperbolic metric outside N r(1−t)(Γ),and on N r(1−t)(Γ)is conformally equivalent to the hyperbolic metric, as follows.Letφ:[0,1]→[0,1]be a C∞bump function,which is equal to1on the interval[1/3,2/3],which is equal to0on the intervals[0,1/4]and[3/4,1],and which is strictly increasing on[1/4,1/3]and strictly decreasing on[2/3,3/4].Then define the ratiog t length elementr(1−t) We are really only interested in the behaviour of the metrics g t as t→1.As such,the choice of r is irrelevant.However,for convenience,we willfix some small r throughout the remainder of§1.The deformed metrics g t have the following properties:Lemma1.18(Metric properties).The g t metric satisfies the following properties:(1)For each t there is an f(t)satisfying r(1−t)/4<f(t)<3r(1−t)/4such thatthe union of tori∂N f(t)(Γ)are totally geodesic for the g t metric(2)For each componentγi and each t,the metric g t restricted to N r(γi)admits afamily of isometries which preserveγi and acts transitively on the unit normal bundle(in M)toγi(3)The area of a disk cross–section on N r(1−t)is O((1−t)2).(4)The metric g t dominates the hyperbolic metric on2–planes.That is,for all2–vectorsν,the g t area ofνis at least as large as the hyperbolic area ofνProof.Statement(2)follows from the fact that the definition of g t has the desired symmetries.Statements(3)and(4)follow from the fact that the ratio of the g t metric to the hyperbolic metric is pinched between1and3.Now,a radially sym-metric circle linkingΓof radius s has length2πcosh(s)in the hyperbolic metric, and therefore has length2πcosh(s)(1+2φ(s/r(1−t)))10DANNY CALEGARI AND DAVID GABAIin the g t metric.For sufficiently small(butfixed)r,this function of s has a local minimum on the interval[r(1−t)/4,3r(1−t)/4].It follows that the family of radially symmetric tori linking a component ofΓhas a local minimum for area in the interval[r(1−t)/4,3r(1−t)/4].By property(2),such a torus must be totally geodesic for the g t metric. Notation1.19.We denote length of an arcα:I→M with respect to the g t metricas lengtht (α(I)),and area of a surfaceψ:R→M with respect to the g t metric asarea t(ψ(R)).1.5.Constructing the homotopy.As afirst approximation,we wish to construct surfaces in M\Γwhich are globally least area with respect to the g t metric.There are various tools for constructing least area surfaces in Riemannian3-manifolds under various conditions,and subject to various constraints.Typically one works in closed3-manifolds,but if one wants to work in3-manifolds with boundary,the “correct”boundary condition to impose is mean convexity.A co-oriented surface in a Riemannian3-manifold is said to be mean convex if the mean curvature vec-tor of the surface always points to the negative side of the surface,where it does not vanish.Totally geodesic surfaces and other minimal surfaces are examples of mean convex surfaces,with respect to any co-orientation.Such surfaces act as bar-riers for minimal surfaces,in the following sense:suppose that S1is a mean convex surface,and S2is a minimal surface.Suppose further that S2is on the negative side of S1.Then if S2and S1are tangent,they are equal.One should stress that this barrier property is local.See[MSY]for a more thorough discussion of barrier surfaces.Lemma1.20(Minimal surface exists).Let M,Γ,S be as in the statement of Theo-rem1.10.Let f(t)be as in Lemma1.18,so that∂N f(t)(Γ)is totally geodesic with re-spect to the g t metric.Then for each t,there exists an embedded surface S t isotopic in M\N f(t)(Γ)to S,and which is globally g t–least area among all such surfaces.Proof.Note that with respect to the g t metrics,the surfaces∂N f(t)(Γ)described in Lemma1.18are totally geodesic,and therefore act as barrier surfaces.We remove the tubular neighborhoods ofΓbounded by these totally geodesic surfaces,and denote the result M\N f(t)(Γ)by M′throughout the remainder of this proof.We assume,after a small isotopy if necessary,that S does not intersect N f(t)for any t,and therefore we can(and do)think of S as a surface in M′.Notice that M′is a complete Riemannian manifold with totally geodesic boundary.We will construct the surface S t in M′,in the same isotopy class as S(also in M′).If there exists a lower bound on the injectivity radius in M′with respect to the g t metric,then the main theorem of[MSY]implies that either such a globally least area surface S t can be found,or S is the boundary of a twisted I–bundle over a closed surface in M′,or else S can be homotoped off every compact set in M′.First we show that these last two possibilities cannot occur.If S is nonseparating in M,then it intersects some essential loopβwith algebraic intersection number 1.It follows that S cannot be homotoped offβ,and does not bound an I–bundle. Similarly,ifγ1,γ2are distinct geodesics ofΓseparated from each other by S,then theγi’s can be joined by an arcαwhich has algebraic intersection number1with the surface S.The same is true of any S′homotopic to S;it follows that S cannot be homotoped off the arcα,nor does it bound an I–bundle disjoint fromΓ,and therefore does not bound an I–bundle in M′.SHRINKWRAPPING AND THE TAMING OF HYPERBOLIC3–MANIFOLDS11 Now suppose that the injectivity radius on M′is not bounded below.We use the following trick.Let g′t be obtained from the metric g t by perturbing it on the complement of some enormous compact region E so that it has aflaring end there, and such that there is a barrier g′t-minimal surface close to∂E,separating the com-plement of E in M′from S.Then by[MSY]there is a globally g′t least area surface S′t,contained in the compact subset of M′bounded by this barrier surface.Since S′t must either intersectβorα,by the Bounded Diameter Lemma1.15,unless the hyperbolic area of S′t∩E is very large,the diameter of S′t in E is much smaller than the distance fromαorβto∂E.Since by hypothesis,S′t is least area for the g′t metric,its restriction to E has hyperbolic area less than the hyperbolic area of S, and therefore there is an a priori upper bound on its diameter in E.By choosing E big enough,we see that S′t is contained in the interior of E,where g t and g′t agree. Thus S′t is globally least area for the g t metric in M′,and therefore S t=S′t exists for any t.The bounded diameter lemma easily implies the following:Lemma1.21(Compact set).There is afixed compact set E⊂M such that the surfaces S t constructed in Lemma1.20are all contained in E.Proof.Since the hyperbolic areas of the S t are all uniformly bounded(by e.g. the hyperbolic area of S)and are2–incompressible rel.Γ,they have uniformly bounded diameter away fromΓoutside of Margulis tubes.Since for homological reasons they must intersect the compact setsαorβ,they can intersect at most finitely many Margulis tubes.It follows that they are all contained in afixed bounded neighborhood E ofαorβ,containingΓ.To extract good limits of sequences of minimal surfaces,one generally needs a priori bounds on the area and the total curvature of the limiting surfaces.Here for a surface S,the total curvature of S is just the integral of the absolute value of the(Gauss)curvature over S.For minimal surfaces of afixed topological type in a manifold with sectional curvature bounded above,a curvature bound follows from an area bound by Gauss–Bonnet.However,our surfaces S t are minimal with respect to the g t metrics,which have no uniform upper bound on their sectional curvature,so we must work slightly harder to show that the the S t have uniformly bounded total curvature.More precisely,we show that their restrictions to the complement of anyfixed tubular neighborhood Nǫ(Γ)have uniformly bounded total curvature.Lemma1.22(Finite total curvature).Let S t be the surfaces constructed in Lemma1.20. Fix some small,positiveǫ.Then the subsurfacesS′t:=S t∩M\Nǫ(Γ)have uniformly bounded total curvature.Proof.Having chosenǫ,we choose t large enough so that r(1−t)<ǫ/2.Observefirstly that each S t has g t area less that the g t area of S,and therefore hyperbolic area less that the hyperbolic area of S for sufficiently large t.Letτt,s=S t∩∂N s(Γ)for small s.By the coarea formula(see[Fed],[CM]page 8)we can estimatearea(S t∩(Nǫ(Γ)\Nǫ/2(Γ)))≥ ǫǫ/2length(τt,s)ds12DANNY CALEGARI AND DAVID GABAIIf the integral of geodesic curvature along a componentσofτt,ǫis large,then the length of the curves obtained by isotopingσinto S t∩Nǫ(Γ)grows very rapidly, by the definition of geodesic curvature.Since there is an a priori bound on the hyperbolic area of S t,it follows that there cannot be any long components ofτt,s with big integral geodesic curvature.More precisely,consider a long componentσofτt,s.For l∈[0,ǫ/2]the boundaryσl of the l-neighborhood ofσin S t∩Nǫ(Γ)is contained in Nǫ(Γ)\Nǫ/2(Γ).If the integral of the geodesic curvature alongσl were sufficiently large for every l,then the derivative of the length of theσl would be large for every l,and therefore the lengths of theσl would be large for all l∈[ǫ/4,ǫ/2].It follows that the hyperbolic area of theǫ/2collar neighborhood ofσin S t would be very large,contrary to existence of an a priori upper bound on the total hyperbolic area of S t.This contradiction implies that for some l,the integral of the geodesic curvature alongσl can be bounded from above.To summarize,for each constant C1>0 there is a constant C2>0,such that for each componentσofτt,ǫwhich has length ≥C1there is a loopσ′⊂S t∩(Nǫ(Γ)\Nǫ/2(Γ))isotopic toσby a short isotopy,satisfyingσ′κdl≤C2On the other hand,since S t is g t minimal,there is a constant C1>0such that each componentσofτt,ǫwhich has length≤C1bounds a hyperbolic globally least area disk which is contained in M\Nǫ/2(Γ).For t sufficiently close to1,such a disk is contained in M\N r(1−t)(Γ),and therefore must actually be a subdisk of S t.By the coarea formula above,we can chooseǫso that length(τt,s)is a priori bounded.It follows that if S′′t is the subsurface of S t bounded by the compo-nents ofτt,s of length>C1then we have a priori upper bounds on the area of S′′t, on ∂S′′tκdl,and on−χ(S′′t).Moreover,S′′t is contained in M\N r(1−t)where the metric g t agrees with the hyperbolic metric,so the curvature K of S′′t is bounded above by−1pointwise,by Lemma1.2.By the Gauss–Bonnet formula,this gives an a priori upper bound on the total curvature of S′′t,and therefore on S′t⊂S′′t. Remark1.23.A more highbrow proof of Lemma1.22follows from Theorem1 of[S],using the fact that the surfaces S′t are locally least area for the hyperbolic metric,for t sufficiently close to1(depending onǫ).Lemma1.24(Limit exists).Let S t be the surfaces constructed in Lemma1.20.Then there is an increasing sequence0<t1<t2<···such that lim i→∞t i=1,and the S tconverge on compact subsets of M\Γin the C∞itopology to some T′⊂M\Γwith closure T in M.Proof.By definition,the surfaces S t have g t area bounded above by the g t area of S.Moreover,since S is disjoint fromΓ,for sufficiently large t,the g t area of S is equal to the hyperbolic area of S.Since the g t area dominates the hyperbolic area, it follows that the S t have hyperbolic area bounded above,and by Lemma1.22, for anyǫ,the restrictions of S t to M\Nǫ(Γ)have uniformly boundedfinite total curvature.。

几何英语知识点归纳总结In this article, we will delve into the key concepts and principles of geometry, covering a wide range of topics from basic shapes to advanced theorems. By the end of this article, you will have a comprehensive understanding of geometry and be able to apply its principles to solve a variety of problems.Basic Concepts in Geometry1. Points, Lines, and Planes: The foundation of geometry lies in the ideas of points, lines, and planes. A point is a location in space, represented by a dot. A line is a straight path that extends in both directions infinitely, with no width or thickness. A plane is a flat, two-dimensional surface that extends infinitely in all directions.2. Angles: An angle is formed when two rays share a common endpoint, referred to as the vertex. Angles are measured in degrees, with a full circle representing 360 degrees. There are different types of angles, including acute angles (less than 90 degrees), obtuse angles (greater than 90 degrees), right angles (exactly 90 degrees), straight angles (exactly 180 degrees), and reflex angles (greater than 180 degrees).3. Polygons: A polygon is a closed shape made up of straight line segments. The most common types of polygons are triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), hexagons (6 sides), and so on. Polygons can be classified based on the number of sides and angles they have.4. Circles: A circle is a set of all points in a plane that are equidistant from a given center point. The distance from the center to any point on the circle is called the radius, and the distance across the circle through the center is called the diameter. The ratio of the circumference of a circle to its diameter is a constant value known as pi (π), approximately equal to 3.14159.5. Similarity and Congruence: Two geometric figures are similar if they have the same shape but different sizes. They are congruent if they have the same shape and size. These concepts are fundamental in understanding the relationships between different geometric figures.6. Perimeter and Area: The perimeter of a shape is the distance around its boundary, while the area is the measure of the space inside the boundary. Different formulas are used to calculate the perimeter and area of various shapes, such as rectangles, triangles, circles, and so on.Advanced Concepts in Geometry1. Pythagorean Theorem: This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It is expressed as a^2 + b^2 = c^2, where a and b are the lengths of the two shorter sides, and c is the length of the hypotenuse.2. Theorems of Euclidean Geometry: Euclidean geometry, named after the ancient Greek mathematician Euclid, is the study of plane and solid figures based on a set of axioms and theorems. Some of the key theorems include the Parallel Postulate, the Angle Sum Theorem, the Pythagorean Theorem, and the Midpoint Theorem.3. Transformations: Transformations in geometry refer to the ways in which a figure can be moved, reflected, rotated, or scaled without changing its shape or size. Common transformations include translations (sliding), reflections (flipping), rotations (turning), and dilations (resizing).4. Coordinates and Graphs: The coordinate plane is a fundamental tool in geometry, consisting of two perpendicular number lines that intersect at the origin (0,0). Points on the plane are represented by ordered pairs of numbers (x,y) called coordinates. By plotting points on the coordinate plane, geometric figures and relationships can be visualized and analyzed.5. Trigonometry: Trigonometry is the branch of mathematics that deals with the study of angles and the lengths of their sides in triangles. It is an essential tool in understanding the relationships between angles and sides, and is widely used in fields such as engineering, physics, and navigation.6. Three-Dimensional Geometry: In addition to the two-dimensional shapes and figures, geometry also encompasses three-dimensional objects such as prisms, pyramids, spheres, cones, and cylinders. Three-dimensional geometry involves the measurement of volume, surface area, and spatial relationships between solid figures.Applications of GeometryThe principles of geometry have a wide range of applications in various fields of study and professions. Some of the key applications include:1. Architecture: Architects use geometry to design and construct buildings, bridges, and other structures. Understanding principles of symmetry, proportion, and spatial relationships is crucial in creating aesthetically pleasing and structurally sound designs.2. Engineering: Engineers utilize geometry in designing and analyzing mechanical components, electrical circuits, and structural frameworks. Geometric concepts such as vectors, forces, and dimensions play a critical role in the field of engineering.3. Cartography: Cartographers use geometry in creating maps and geographic information systems. By understanding the principles of projections, scales, and angles, accurate representations of geographic features can be produced.4. Art and Design: Artists and designers often incorporate geometric shapes, patterns, and proportions in their work. Understanding geometric principles allows them to create compositions with balance, harmony, and visual appeal.5. Computer Graphics: The field of computer graphics heavily relies on geometric algorithms and principles to create visual representations of virtual environments, objects, and characters in video games, movies, and simulations.ConclusionGeometry is a fundamental branch of mathematics that encompasses the study of shapes, sizes, and properties of space. By understanding the basic concepts of points, lines, angles, polygons, and circles, as well as the advanced principles of transformations, coordinates, trigonometry, and three-dimensional geometry, we are able to make sense of the world around us in a more precise and systematic manner.The principles of geometry have a wide range of applications in various fields of study and professions, including architecture, engineering, cartography, art and design, and computer graphics. By applying the principles of geometry to solve problems and analyze real-world scenarios, we are able to create, innovate, and understand the world in a more profound way. As we continue to advance in our understanding of geometry, we open up new possibilities for exploration, discovery, and creativity in the world of mathematics and beyond.。

a r X i v :0802.1490v 1 [h e p -l a t ] 11 F eb 2008Closed k-strings in SU(N)gauge theories :2+1dimensionsBarak Bringoltz a and Michael Teper b a Department of Physics,University of Washington,Seattle,WA 98195-1560,USA b Rudolf Peierls Centre for Theoretical Physics,University of Oxford,1Keble Road,Oxford OX13NP,UK Abstract We calculate the ground state energies of closed k -strings in (2+1)-dimensional SU(N)gauge theories,for N =4,5,6,8and k =2,3,4.From the dependence of the ground state energy on the string length,we infer that such k -strings are described by an effective string theory that is in the same bosonic universality class (Nambu-Goto)as the fundamental string.When we compare the continuum k -string tensions to the corresponding fundamental string tensions,we find that the ratios are close to,but typically 1%−2%above,the Casimir scaling values favoured by some theoretical approaches.Fitting the N -dependence in a model-independent way favours an expansion in 1/N (as in Casimir scaling)rather than the 1/N 2that is suggested by naive colour counting.We also observe that the low-lying spectrum of k -string states falls into sectors that belong to particular irreducible representations of SU (N ),demonstrating that the dynamics of string binding knows about the full gauge group and not just about its centre.1IntroductionIn this paper we calculate the ground state energy of a confiningflux loop that winds once around a spatial torus of length l,with theflux in a higher representation than the funda-mental.We do so for a number of SU(N)groups in D=2+1dimensions.This study complements recent calculations for fundamentalflux loops[1]and the excited state spectra of these[2].In a forthcoming paper we shall analyse the excited state spectrum of multiply woundflux loops[3]and similar calculations in D=3+1dimensions are under way.These numerical calculations have a number of motivations.Firstly there are old ideas that the relevant degrees of freedom of linearly confining SU(N)gauge theories are string-likeflux tubes and that these should be described by an effective string theory[4]which might describe all the important physics of thefield theory,particularly as N→∞[5].Calculations such as ours can serve to test this programme;and,at the very least,to pin down the details of the effective string theory that describes the dynamics of longflux tubes[6,7].Secondly,there has been dramatic recent progress in the construction of string duals to various supersymmetric field theories[8].There is a great effort to extend this to gauge theories and to QCD(see [9]for recent reviews).Our calculations may provide useful information in the search for the appropriate construction.Our recent calculation[1,2]of the spectrum of a closed loop of fundamentalflux as a function of its length l,demonstrated that this spectrum is accurately given by the simple Nambu-Goto string model[10]down to lengths much smaller than those at which an expansion in powers of1/σl2diverges.While this does not contradict the conventional approach of starting with a background consisting of a long‘straight’string and expanding influctuations around it[6],it does imply that one can do considerably better by expanding instead in small corrections around the full Nambu-Goto solution.Hints of this possibility can already be discerned in some recent work,[11,12],based on[6,7]respectively,that showed that the next term beyond the Luscher term[6]in the expansion in powers of1/σl2is also universal and has the same coefficient as in the Nambu-Goto case.This fact,that we can identify the states as being essentially those of the Nambu-Goto model with the corresponding‘phonon’occupation numbers,means that the observed small deviations from Nambu-Goto have the potential to provide detailed and powerful constraints on the form of the additional string interactions that are needed.The k-string that is the object of our study in this paper,can be thought of as a bound state of k fundamental strings.Earlier calculations have shown that for N<∞there do indeed exist such bound states both in D=2+1and in D=3+1[13,14,15,16].This binding cannot be readily incorporated into the usual analytic frameworks[6,7]since the binding interaction presumably involves the exchange of small closed loops of string that are not under analytic control.Thus such strings are of particular interest.In the case of SU(N)gauge theories in D=2+1there is an additional motivation.There has been significant recent progress towards an analytic solution of these theories using a variational Hamiltonian approach[17].(See also related work in[18].)These calculations make predictions for both the fundamental string tension and for k-string tensions.In[1]we found that the former prediction is very accurate and improves with increasing N,being only1about1%offthe correct value at N=∞.(Although this discrepancy is statistically very significant.)In this paper we shall provide a similar test for the predicted k-string tensions.In the next section we describe our method for calculating the energy of a closed k-string. We then present,in Section3,our most accurate results for the l-dependence of this energy (which happens to be for the case k=2)and compare to the expectation of the simple bosonic string model.In Section4we calculate the k-string tensions,extrapolate these to the continuum limit,and test the conjecture that they scale with the appropriate Casimir. We follow this,in Section5,with a model-independent analysis of the N-dependence of these string tensions.Finally,in Section6,we look at the wave-functions of the ground and excited states and show that in fact they fall into representations of the group,rather than being determined simply by the center.A summary of this work has been presented in[19].2MethodologyConsider a spatial torus of length l with all the other tori so large that they may be considered infinite.If the system is linearly confining,anyflux around the torus will be confined to‘string-like’flux tubes.If the source from which such aflux would emanate transforms asψ→z kψunder a gauge transformation,z,that belongs to the center,Z N,of the SU(N)gauge group, theflux tube is called a k-string.This is a useful categorisation since it is invariant under screening by gluons.For large l the ground state k-string energy E0,k(l)gives us the k-string tension,E0,k(l)σk=liml→∞lower N potentials calculated in[21]points to the possibility of an interesting answer to this question.Possible effective string theories fall into universality classes.If we expand E0(l)in powers of1/σl2aroundσl,then for a bosonic string,where the only zero modes are those that arise from the D−2=1transverse translations,thefirst two correction terms are universal[11,12]E0(l)=σl−cπ72σl3+...;c=1.(2)More generally,if there are additional zero modes along the string,then we may have c=1 (although in that case only thefirst correction term has been shown to be universal).The simplest example of a bosonic string is the Nambu-Goto free string model,whose ground state energy is[10]E0(l)=σl 1−πσl2 1NReTr U p};β=2Nis still relatively small.This can be achieved using standard smearing/blocking/variational techniques(see e.g.[22,14]).If we denote a Polyakov loop in the fundamental k=1represen-tation by l p,then typical operators for k-strings will beφl=Tr{l k−jp}Tr{l p}j and we include all such operators in our variational basis,including their smeared/blocked versions.Since E0(l)becomes large for large l,it can only be accurately calculated for modest values √σand so we need to know what are the importantfinite l corrections to the linear term, of lE0(l)≃σl if we are to extract a reliable value forσ.This is even more important for heavier strings with k>1,since these are heavier than the fundamental k=1string.In[1]it was shown that for the fundamental string,corrections to eqn(3)are extremely small,even very close to l c and one can safely extrapolate using eqn(3)from,say,l√σf∼3.However there is no guarantee that this will continue to be the case for k>1.Indeed we expect that atfixed k the binding will vanish as N→∞,and so in that limit,at anyfixed l however large,the lightest k-string will actually be k fundamental strings,each satisfying eqn(2)[23].Moreover, the fact that all the strings must share the same critical length,l c,leads us to expect much larger deviations from eqn(3)at small l for k>1.In addition there have been suggestions [24]that k>1strings do not belong to the bosonic string universality class and have c>1. Thus it is important to determine the l-dependence of the ground state k-string and this is the issue to which we now turn.3Central charge and l dependence of k-stringsWe have performedfinite volume studies for N=2,3,4,5,6,8at various values of the lattice spacing a.Here we shall present two of our most accurate studies for k=2strings.These are for SU(4)atβ=32.0and SU(5)atβ=80.0.In units of the fundamental string tension, the lattice spacings correspond to a√σf≃0.2153and a√σf≃0.1299respectively.In Table1we present the k=2ground state energy as a function of l=aL x.We also present,for comparison,the corresponding energy of the fundamental k=1string.The critical value of l at which one loses the string is given by l c∼5a and l c∼8a for SU(4) and SU(5)respectively.Thus our range of closed string lengths extends down to nearly the minimimum possible value.There are many useful ways to analyse the l-dependence.Here we shallfit neighbouring values of l to a Nambu-Goto formula with an‘effective’central charge:E0(l)=σl 1−c effπσl2 1see from eqn(6),it does mean that the scale in fixed physical units is slightly different for the k =1and k =2analyses.The horizontal error bars end at the two values of l used in the fits to eqn(6).In Fig.2we have a corresponding plot for SU(5).We observe in Figs.1,2quite strong evidence for the claim that at long distances the k =2flux tube behaves like a simple bosonic string,just like the fundamental flux tube.This contradicts the conjecture in [24]that lim l →∞c eff =σk /σf .(This latter ratio is,from our fits,about 1.35and 1.52for SU(4)and SU(5)respectively.)However it is clear that the corrections to Nambu-Goto are very much larger for k =2than for k =1.Indeed,if we limited our analysis to l √σk <2.5we might be led to agree with the conjecture of [24].One way to quantify the size of the deviation is to add to eqn(3)a correction term that is of higher order than the universal terms in eqn(2),i.e.E 0(l )=σl 1−πσl 2+c ′1σl )51For a given k the smallest Casimir arises for the totally antisymmetric representation,and this should therefore provide the ground state k-string tension:σk.(9)N−1This is the part of the‘Casimir Scaling’hypothesis that we shall be mainly testing in this paper.For this purpose it is useful to have an alternative conjecture that possesses the correct general properties.A convenient and well-known example is provided by the trigonometric formσkN(10)Nthat was originally suggested on the basis of an M-theory approach to QCD[27].In fact the full prediction of[17]forσk is more specific than eqn(9),since it also predicts a value forσf in terms of g2,and including this gives:1σkN2statistical errors that are typical of the present calculation).We therefore use eqn(7)to extract values ofσk from the corresponding E0(l).Having obtained values ofσk at various values ofβwe now want to extrapolate the dimensionless ratiosσk/σf to the continuum limit.We do so using:σk(a)+c1a2σf+c2a4σ2f+ (12)σf(0)where the c i can be treated as constants over our range ofβ.Typically extrapolations of this kind include just the O(a2)correction,and one drops the points at the largest values of a until a statistically acceptablefit is possible.We shall also perform a O(a4)fit so as to estimate the systematic error in the continuum extrapolation.We show the resulting values ofσk/σf in SU(5)in Figs.3,4,where the calculations have been performed using single and double exponentialfits respectively.We also show linear O(a2)and quadratic O(a4)extrapolations to the continuum limit in each case.The resulting continuum values,together with the values ofχ2per degree of freedom for the bestfits,are listed in paring Fig.3with Fig.4we see that the S and D values are quite consistent although the errors of the latter are significantly larger.The corresponding contin-uum limits are,as a result,also very similar.If we now compare the two different continuum extrapolations,we see that including an extra O(a4)correction reduces the continuum value by a little more than one standard deviation.In principle thefit with the extra correction term should be more reliable.However the coefficient of such a term is in practice largely determined by the calculations at the largest values of a,where there is always the danger that one is being influenced by the strong-to-weak coupling crossover where the power series expansion in a2breaks down.In any case,we see from this study that neither the use of double exponentialfits,nor the inclusion of higher order terms in the continuum extrapola-tion,makes much difference to thefinal result.Our qualitative conclusion is that while the continuum ratio is close to the Casimir Scaling value of1.5,there is a significant and robust discrepancy at the∼1.5%level.We have repeated the above calculation for SU(4),SU(6)and SU(8)and the resulting continuum values,using O(a2)extrapolations,are presented in Table3.Thefirst error is statistical and the second is intended to give some idea of the maximum possible systematic error,by looking at what one obtains from various combinations of S and Dfits toσk and σf.(As such it is not an estimate of the actual systematic error which would undoubtedly be significantly smaller.)Some remarks.Firstly,for SU(6)our bestfit has a rather poor, but not impossible,χ2.We have therefore doubled the errors so as to achieve a reasonable confidence level.This will ensure that the N=6value does not distort our analysis of the N-dependence in the next Section.Second remark:for N=8one of the twofinite volume calculations has a very poorχ2when we attempt to extract the string tensions.We therefore do not include the resulting values,that have very small errors but very small confidence levels,in our continuum extrapolations,but instead take from that study the string tension as calculated from the lattice whose size(in physical units)is the same as we use at other values ofβ.(Some freedom as to which values to include at thoseβat which we performfinite volume studies,as well as some freedom in thefitting process,explains why the SU(5)values7in Table 3do not coincide exactly with those in Table 2.)Note that as k increases,the loop energy also increases,and the errors become less reliable.Nonetheless,despite these caveats,we see from Table 3that we can confidently conclude that while the string tension ratios are remarkably close to the Casimir Scaling values,and much further away from (MQCD)Sine Scaling,there is a definite discrepancy that is typically at the 1−2%level.It is interesting to note that if we take the actual values of √σf /g 2N as obtained in [1]together with the values of σk in eqn(11),we obtain the values of σk /σf listed in the last column of Table 3.We see that these values are largely consistent with our calculated numbers.Thatis to say,while the prediction of [17]for σf is too high by ∼1−2%,the prediction for √σk /g 2Nwith k >1is actually consistent with the values we obtain.This is particularly significant for the k =2values which are the ones that we determine most accurately and reliably.5N -dependenceAn interesting feature of the Casimir Scaling hypothesis in eqn(9)is that if we expand the expression for σk /σf in powers of 1/N at fixed k ,we see that the leading correction is O (1/N )rather than the O (1/N 2)that one would expect from the usual colour counting rules.This observation [28]has generated some controversy [29,16].It is therefore interesting to see if we can learn something about the power of the leading correction from a more model-independent analysis of our results.We begin with our results for the k =2continuum string tensions,as these are more accurate than those for larger k .We attempt a fit with the ansatzσk =2N p −bpossibly a large systematic error as well.Proceeding anyway,wefit with the ansatzσk=N/22 a+bThere is some evidence from calculations of potentials between sources in different repre-sentations that the correspondingfluxes do formflux tubes[26],but such calculations do not go to large enough separations to be unambiguous.There is also some evidence from earlier calculations of closed loops,that the lowest excited k=2strings fall approximately into symmetric and anti-symmetric representations[13],although the observed near-degeneracies in these states[14]have been interpreted as saying that these are in fact the same states, and that the categorisation at a given k into different representions is illusory:that is to say, confining strings only know about the center of the group[30].To determine what is actually the case,one needs accurate calculations of a significant number of excited states at each k.This requires a large basis of operators enabling one to obtain good overlaps onto these excited states,paralleling the k=1calculations in[1].Such a calculation is under way[3].In the meantime,it is interesting to see what we can extract from the present calculation which has a rather limited basis,with overlaps on to the excited k-string states that are only moderately good.We restrict ourselves here to the k=2excited state spectrum which is the one that we can determine most accurately.The operators whose correlators we have calculated are of the form Tr l2and{Tr l}2where l is a Polyakov loop.There will be such operators corresponding to each of the smearing/blocking levels and this provides us with a non-trivial basis for our variational calculation of the k=2spectrum.This basis of operators allows us to construct Polyakov loops in the totally symmetric(2S)and antisymmetric(2A)representations:Tr2A l=Tr l2−{Tr l}2;Tr2S l=Tr l2+{Tr l}2.(15) To construct k=2operators in other representations would require higher powers,e.g. Tr l3Tr l†,which we have not calculated here.Thus we shall limit our analysis to the k=2A and k=2S ing the basic operators in eqn(15),we construct p=0operators at time t by summing up Tr2A,2S l over the spatial coordinates.We call the resulting operators Φ2A(t)andΦ2S(t)where there is an additional suppressed label,b,for the blocking level of the gauge links used in the construction of l.These are our basis operators and our variational procedure will(ideally)produce the linear combinations that are closest to the actual energy eigenstates.One can immediately learn something interesting from these basis operators.Consider the normalised overlap of a symmetric operatorΦ2S at blocking levels b S with an antisymmetric operatorΦ2A at blocking levels b A:O AS(b A,b S)= Φ†2A(0)Φ2S(0)and ensures that the string is long enough,l √σf≃3.1.Our results show that the overlaps are remarkably small.Indeed they are all consistent with zero,within very small errors,except for those that involve the very largest blocking level.Blocked links at this level spread right across the spatial volume and we have convincing evidence from calculations on larger volumes that the somewhat enhanced overlap is primarily a finite volume effect.Nonetheless even these non-zero values are in fact very small.This provides striking evidence that the screening dynamics is a weak perturbation on the classification of states into representations of SU(N ).We now turn to our variational estimates of the the k =2string eigenstates,ΨJ ,that we obtain using the full basis of Φ2A (t )and Φ2S (t )operators.We project the resulting eigenstates onto the k =2A and k =2S subspaces.(Which we do by forming an orthogonal basis out of our non-orthogonal Φ2A,2S operators.)We call the corresponding overlaps O JA and O JS ,where we expect |O JA|2+|O JS|2≃1since the k =2A and k =2S subspaces are nearly orthogonal.If the subspaces were exactly orthogonal then it would of course immediately follow that theeigenstates would fall into exact k =2A and k =2S representations.What we find is that they appear to do so to quite a good approximation.An example is shown in Fig.7.Here we plot the overlaps |O JA|2,|O JS |2for the first five k =2states,J =1,...,5,as a function of thestring length l ,in a calculation in SU(6)at β=59.4(which corresponds to a √σf≃0.28).What we see is that the states are almost entirely either pure k =2A or k =2S except at certain values of l where the state changes from one representation to the other.Since this always happens to pairs of states at the same value of l and involves an opposite change in representation in this pair,it is clear what is happening:as we increase l thetwo energy levelscross at that l .So,for example,in Fig.7the states that are Ψ2and Ψ3for l √σf <3exchange their ordering in energy for l √σf >4.And a similar exchange occurs for Ψ4and Ψ5,but at a larger value of l .What we see,therefore,is that the k -string states belong to particular SU(N )representations to a very good approximation.The apparent near-degenaracies observed in[14]are in fact accidental:they arise from the (natural)fact that the various excited states have energies that vary differently with l ,so that they cross as l increases and when they do so we have degeneracies.As we have just seen,the first k =2crossing occurs at l √σf∼3.5which is precisely where such degeneracies have been previously noted to occur [13,14].It is unambiguous that what we are seeing here is not a single state,as suggested in [30],but rather two that are nearly degenerate at appropriate values of l .If we focus on the lightest three states in Fig.7,we see that the ground state is always totallyanti-symmetric,while the first excited state is initially symmetric and then,for l √σf >4,antisymmetric.This very different dependence on l between the ground state k =2S state and the first excited k =2A state is easily understood if we think of there being two separate Nambu-Goto towers of states labelled by the k =2A and k =2S representations respectively.In the Nambu-Goto model the energy levels would be given byE n (l )=σR l 1+8ππ24 1E 0(l )increases approximately linearly with l in the range of l relevant here.On the other hand the first excited state with n =1will clearly have a very different variation with l .This is illustrated in Fig.8where we plot the Nambu-Goto energy levels (divided by σf l )for towers of states in the k =2A and k =2S representations of SU(6)using the string tensions that are predicted by Casimir Scaling.One sees how the very different l -dependence of the ground state k =2S string and the first excited k =2A string ensures that they cross for a value of l √σf close to where we observe the crossing in Fig.7.Our estimates of the actual energy levels are very roughly consistent with such a scenerio,although a reliable and precise comparison must await a dedicated study of the k >1energy spectrum [3].In the meanwhile what we have is very good evidence that the k -string states fall,to a good approximation,into representations of SU(N ),and strong indications,from the crossings of the states,that there exists a ground state flux tube in representations such as k =2S where it is not completely stable.7ConclusionsIn this paper we have studied k -strings that wind around a spatial torus of length l .This setup is convenient because it involves no sources so one knows that,for l ≥1/T c ,all that one has is a closed string-like flux tube.It provides a clean way to determine the effective string theory describing the flux tube.Once this question has been addressed the second,equally important question of how the effective string theory at long distances matches onto short-distance asymptotic freedom,can be conveniently addressed through the calculation of the potential between sources as a function of the distance between these sources.For example,one can ask whether the running coupling in the ‘potential scheme’V (r )≡−C αv (r )breakdown of Casimir Scaling that we observe can be entirely attributed to the fact that the predicted value ofσf in[17]is slightly higher than its actual value as obtained in[1].One should perhaps not read too much significance into this agreement.The important point is that any discrepancy between the predictions forσk in[17]and the true values are no larger for k>1strings than they are for the fundamental string.•An analysis of how the k=2string tension varies with N strongly suggests that the corrections to the N=∞limit,σk=2=2σf,come in powers of1/N(as in Casimir Scaling) rather than in powers of1/N2,as suggested by standard colour counting arguments.There is also some statistically weaker evidence for this from our analysis of the N-dependence of σk=N/2for N=4,6,8.•The k-string eigenstates fall,very accurately,into representations of SU(N).That is to say,k-strings know about the full group representation of theflux,and not just about its transformation properties under the centre.The near-degeneracies previously observed be-tween some states assigned to different representations,should be interpreted as level crossing (as a function of the string length l).The intriguing possibility that stable k-strings(and per-haps even unstable‘resonant’strings)may possess their own towers of Nambu-Goto states, receives some support from preliminary precision results on the k-string spectrum[3].AcknowledgementsWe acknowledge useful discussions with Ofer Aharony,Nick Dorey,James Drummond,Pietro Giudice,JeffGreensite,Dimitra Karabali,V.P.Nair,David Tong,Peter Weisz and,in partic-ular,with Andreas Athenodorou,with whom we are collaborating on a closely related project. 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a r X i v :d g -g a /9710009v 1 9 O c t 1997Cocycles,symplectic structures and intersectionUrsula Hamenst¨a dt1.IntroductionIn the paper [B-C-G]Besson,Courtois and Gallot proved the following remarkable theorem:Let S be a closed rank 1locally symmetric space of noncompact type and let M be a closed manifold of negative curvature which is homotopy equivalent to S .If S and M have the same volume and the same volume entropies (i.e.the same asymptotic growth rate of volumes of balls in their universal covers),then S and M are isometric.One application of this result is the solution of the so-called conjugacy problem for locally symmetric manifolds.This problem can be stated in the following way:The marked length spectrum of a closed negatively curved manifold (M,g )is the function ρM which assigns to each conjugacy class in the fundamental group π1(M )of M the length of the unique g -geodesic representing the class.Two homotopy equivalent negatively curved manifolds M ,N have the same marked length spectrum if there is an isomorphism Ψ:π1(M )→π1(N )such that ρN ◦Ψ=ρM .This is equivalent to the existence of a continuous time preserving conjugacy for the geodesic flows for M and N ([H1]).Such a conjugacy is defined to be a homeomorphism Λof the unit tangent bundle T 1M of M onto the unit tangent bundle T 1N of N which is equivariant under the action of the geodesic flows Φt on T 1M and T 1N ([H1]).The conjugacy problem now asks whether a composition of Λwith the time-t-map of the geodesic flow on T 1N for a suitable t ∈R in the restriction to T 1M of the differential of an isometry of M onto N .This is known to be true for surfaces ([O]).Since the volume entropy of a closed negatively curved manifold equals the topological entropy of the geodesic flow,two manifolds with time preserving conjugate geodesic flows have the same volume entropy.Let X 0be the geodesic spray ,i.e.the generator of the geodesic flow Φt .Recall that there is a H¨o lder continuous Φt -invariant decomposition T T 1M =R X 0⊕T W ss ⊕T W su ,the so called Anosov splitting ,where T W ss (or T W su )is the tangent bundle of the strong stable (or strong unstable)foliation W ss (or W su )on T 1M .The bundle T W ss ⊕T W su is smooth,and hence we obtain a smooth 1-form ωon T 1M by defining ω(X 0)≡1and ω(T W ss ⊕T W su )=0.This form ωis called the canonical contact form .The differential form ω∧(dω)n −1is the volume form of the so called Sasaki metric on T 1M ,and the total mass of T 1M with respect to this volume form equals the product of the volume of M with the volume of the (n −1)-dimensional unit sphere in R n .The pull-back of the canonical contact form on T 1N under a time preserving conju-gacy Λ:T 1M →T 1N of class C 1is the canonical contact form on T 1M .This implies that M and N have the same volume if their geodesic flows are C 1-time-preserving con-jugate.Therefore the result of Besson,Courtois and Gallot gives a positive answer to theconjugacy problem if one of the manifolds under consideration is locally symmetric and if the conjugacy is assumed to be of class C1.One of the objectives of this note is to remove this regularity assumption on the conjugacy.In Section3we show:Theorem A:If M and N are closed negatively curved manifolds with the same marked length spectrum and if the Anosov splitting of T T1M is of class C1,then M and N have the same volume.Manifolds with strictly1/4-pinched sectional curvature or locally symmetric manifolds have C1Anosov splitting.Thus as a corollary from[B-C-G]we obtain:Corollary:If M has the same marked length spectrum as a closed negatively curved locally symmetric space S,then M and S are isometric.Recall that the space of geodesics G˜M of the universal cover˜M of(M,g)is the quotient of the unit tangent bundle T1˜M of˜M under the action of the geodesicflowΦt.Also G˜M can naturally be identified with∂˜M×∂˜M−∆where∂˜M is the ideal boundary of˜M and∆is the diagonal.The space of geodesics is a smooth manifold,and the differential dωof the canonical contact form projects to a smooth symplectic structure on G˜M.The fundamental groupπ1(M)of M acts on G˜M as a group of symplectomorphisms.The cross ratio of the metric g is a function on G˜M×G˜M.We show in Section3that this function can be viewed as a coarse(or rather integrated)version of the symplectic form dωon G˜M.This enables us to construct the measureλg on G˜M defined by the smooth volume form(dω)n−1in a purely combinatorial way from the cross ratio.The measure λg is usually called the Lebesgue Liouville current of the metric g.Our construction is general,but unfortunately so far we are only able to show that the measure we obtain from it is exactly the Lebesgue Liouville current(rather than some multiple of it)under the additional assumption that the Anosov splitting of T T1M is of class C1.A geodesic current for M is a locallyfinite Borel measure on the space of geodesics which is invariant under the action of the fundamental group and under the exchange of the factors in the product decomposition of G˜M=∂˜M×∂˜M−∆.If M and N are two homotopy equivalent closed negatively curved manifolds,then there is a natural π1(M)=π1(N)-equivariant H¨o lder homeomorphism of∂˜M onto∂˜N which induces an equivariant homoeomorphism of G˜M onto G˜N,in particular the spaces of geodesic currents for M and N are naturally identified.With this identification we show in Section4:Theorem B:Let(M,g),(N,h)be homotopy equivalent closed negatively curved man-ifolds.Ifλg=λh and if the Anosov splittings of T T1M and T T1N are of class C1,then M and N have the same marked length spectrum.Recall that a H¨o lder continuous additive cocycle for the geodesicflowΦt is a H¨o lder continuous functionζ:T1M×R→R which satisfiesζ(v,s+t)=ζ(v,s)+ζ(Φs v,t)for all v∈T1M and all s,t∈R.Two cocyclesζ,ξare called cohomologous if there is a H¨o ldercontinuous functionβon T1M such thatζ(v,t)−ξ(v,t)=β(Φt v)−β(v)for all v∈T1M and all t∈R.We denote by[ζ]the cohomology class ofζ.Let F:T1M→T1M be the flip v→F v=−v;theflip Fζof a cocycleζis the cocycle Fζ(v,t)=ζ(FΦt v,t).Two cocyclesζ,ξare cohomologous if and only if this is true for Fζand Fξ.In other words, F induces an action on cohomology classes of H¨o lder cocycles which we denote again by F.We callζquasi-invariant under theflip if the cocyclesζand Fζare cohomologous,i.e. if[ζ]is afix point for the action of F.Consider now a closed negatively curved surface M.Everyflip invariant H¨o lder co-homology class for the geodesicflow defines a unique cross ratio([H5]),and in the two-dimensional case the defining properties of a cross ratio([H5])show that we can obtain from a cross ratio afinitely additive signed measure on G˜M in a in a natural way.In other words,there is a natural linear map which associates to a H¨o lder cohomology class[ζ]a signed measureνζ.Now the length cocycleℓof the negatively curved manifold(M,g)is defined byℓ(v,t)=t for all v∈T1M and all t∈R.For the length cocycleℓof g on the surface M,νℓis just1/2times the Lebesgue Liouville current of g.On the other hand,if the H¨o lder cohomology class is positive,i.e.if it can be repre-sented by a cocycle¯ζwhich satisfies¯ζ(v,t)>0for all v∈T1M and all t>0,then there is another more classical way to define a projective class of currents on G˜ly,a suitable positive multiple[aζ]of[ζ](here a>0is a constant depending on[ζ])defines a Gibbs equilibrium state which is aπ1(M)-invariant measure on G˜M,determined uniquely up to a constant.This measure is a geodesic current if and only if[ζ]=[Fζ],and if this is satisfied we call it a Gibbs-current ofζand denote its projective class in the space of pro-jectivized currents by[µζ].The assignment[ζ]→[µζ]then is a map from a subset of the projectivization of the space offlip invariant cohomology classes into the projectivization of the space of geodesic currents.Ifℓis the length cocycle of the Riemannian metric g on M,then its Gibbs current is the Bowen-Margulis current of g which corresponds to the measure of maximal entropy for the geodesicflow.Katok showed in[K]that the Lebesgue Liouville current and the Bowen Margulis current of a negatively curved metric on a surface are equivalent if and only if the metric has constant curvature.With our above notation this result can be stated as follows:If ℓis the length cocycle of a metric g of negative curvature on the surface M and ifνℓis equal toµℓup to a constant,then g has constant curvature.In Section2we obtain a generalization of Katok’s result to arbitrary positive H¨o lder cohomology classes.For its formulation denote by[ν]the class of a current in the projec-tivization of the space of geodesic currents.We show:Theorem C:Let M be a closed surface of genus g≥2.If[µℓ]=[νℓ]for a positive flip invariant H¨o lder cohomology classℓ,thenℓis the class of the length cocycle of a metric on M of constant curvature.Before we proceed wefix a few more notations.All our manifolds will be closed and equipped with afixed negatively curved metric.We denote by T1M(or T1˜M)the unit tangent bundle of M(or˜M).There is a naturalπ1(M)-equivariant projectionπ:T1˜M→∂˜M.The pre-image of a pointξ∈∂˜M underπequals the stable manifold in T1˜M of alldirections pointing towardsξ.We let W i(v)be the leaf of the foliation W i containing v (i=ss,su)and write P for the canonical projections T1M→M and T1˜M→˜M.We will need the following facts which were pointed out by Ledrappier([L]):The fundamental groupπ1(M)of M acts naturally on the ideal boundary∂˜M of˜M as a group of homeomorphisms.There is a natural bijection between the space of H¨o lder cocycles for the geodesicflow on T1M and the space of H¨o lder cocycles for the action ofπ1(M)on∂˜M.Such a H¨o lder cocycle for the action ofπ1(M)is a H¨o lder continuous functionsζ0:π1(M)×∂˜M→R which satisfiesζ0(ϕ1ϕ2,ξ)=ζ0(ϕ1,ϕ2ξ)+ζ0(ϕ2,ξ)for ϕ1,ϕ2∈π1(M)andξ∈∂˜M.Two such functionsζ0,η0are cohomologous if there is a H¨o lder continuous functionβ0on∂˜M such thatζ0(ϕ,ξ)−η0(ϕ,ξ)=β0(ϕξ)−β0(ξ)for allϕ∈π1(M)and allξ∈∂˜M.Ifξ(ϕ)is the attractingfixed point for the action of ϕ∈π1(M)on∂˜M,thenζ0(ϕ,ξ(ϕ))is called the period ofζ0atϕ.Two H¨o lder cocycles are cohomologous if and only if they have the same periods.The natural map from H¨o lder cocycles for the geodesicflowΦt to H¨o lder cocycles for the action ofπ1(M)on∂˜M preserves the equivalence relation which defines cohomology. Moreover a H¨o lder class[ζ]forΦt is invariant under theflip if and only if the corresponding class[ζ0]for the action ofπ1(M)satisfiesζ0(ϕ,ξ(ϕ))=ζ0(ϕ−1,ξ(ϕ−1))for allϕ∈π1(M) (all this is discussed in[L]).In particular,the space of H¨o lder classes for the geodesicflow of a negatively curved metric g on M as well as the notion offlip-invariance and positivity do not depend on the choice of the metric.Let M be the space ofΦt-invariant Borel probability measures on T1M.For every H¨o lder cocycleζforΦt and everyµ∈M the integral ζdµcan be defined.This integral only depends on the cohomology class[ζ]ofζ.More precisely,letα:T1M→R be a H¨o lder cocycle and write f(v)=α(v,1)for all v∈T1M.Then f is a H¨o lder continuous function on T1M and defines a H¨o lder cocycleαf by the formulaαf(v,t)= t0f(Φs v)ds. If v∈T1M is a periodic point forΦt of periodτ>0thenαf(v,τ)= τ0f(Φs v)ds= τ0α(Φs v,1)ds= τ0(α(v,s+1)−α(v,s))ds= 10(α(v,τ+s)−α(v,s))ds=α(v,τ)by the cocycle equality forα.But this just means thatαandαf are equivalent (compare[L]).Thus every H¨o lder cohomology class can be represented by a H¨o lder con-tinuous function f,and two functions f and g define the same H¨o lder cohomology class if and only if for every elementηfrom the space M ofΦt-invariant Borel probability mea-sures on T1M we have fdη= gdη(this is the Livshicv theorem for H¨o lder continuous functions).Denote by[f]the cohomology class of the cocycleαf defined by f.Recall that the flip F:T1M→T1M defined by F(v)=−v is a diffeomorphism of T1M which satisfies F◦Φt=Φ−t◦F.Call[f]flip-invariant if[f]can be represented by a function g on T1M which is invariant under F.Again this notion is independent of the choice of a negatively curved metric on M and coincides with the notion used above.Thus for a H¨o lder classζand a measureµ∈M we can define ζdµ= ζ(v,1)dµ(v). Then the pressure pr([ζ])of[ζ]is defined to be the supremum of the values hµ− ζdµwhereµranges through M and hµis the entropy ofµ.If[ζ]is positive,i.e.if[ζ]can be represented by a cocycle¯ζwhich satisfies¯ζ(v,t)>0for all v∈T1M and all t>0,then there is a unique number a>0such that[aζ]is normalized i.e.that pr([aζ])=0.The Gibbs current[µζ]of[ζ]then is the projection of the Gibbs equilibrium state of aζto G˜M.As was shown in[B2],for a surface M there is a natural bilinear form,the so called intersection form,on the compact convex space of geodesic currents for M.This bilinear form is continuous with respect to the weak∗-topology on the space of geodesic currents. The intersection of the Lebesgue Liouville current of a negatively curved metric g with any currentβis just twice the integral of the length cocycle of g with respect toβ.One consequence of Theorem B is the following:For every closed manifold M,the map which assigns to the class of the length cocycle of a strictly1/4-pinched negatively curved metric on M its Lebesgue Liouville current is injective.Thus we can define the intersection between a currentνand the Lebesgue Liouville currentλℓof the length cocycle ℓof such a metric as ℓdν.We conjecture that the class of the length cocycle of every negatively curved metric is determined by its Lebesgue-Liouville current,and that it is possible to extend the above function to a continuous non-symmetric bilinear form defined on the vector space spanned by all Gibbs currents.2.Gibbs currents for surfacesTo begin with,let M be an arbitrary closed Riemannian manifold of negative sectional curvature.Recall that the space of cohomology classes of H¨o lder cocycles for the geodesic flow on T1M is independent of the choice of a metric of negative curvature on M(compare [L]and the introduction).We will only considerflip invariant H¨o lder cohomology classes.The cohomology class [f]defined by a function f on T1M is called positive if it can be represented by a positive H¨o lder function.This notion coincides with the one given in the introduction.Denote by H the space of positiveflip invariant H¨o lder classes.This space carries a natural(non-complete)topology as follows:Recall that a geodesic current for M is aΓ=π1(M)-invariant locallyfinite Borel measure on∂˜M×∂˜M−∆where∆is the diagonal in∂˜M×∂˜M,which in addition is invariant under the natural involution of∂˜M×∂˜M exchanging the two factors.Geodesic currents correspond naturally tofiniteΦt-invariant Borel measures on T1M which are invariant under theflip.We equip the space C of geodesic currents with the weak*-topology. With this topology,C is a locally compact space.For everyα∈H and every currentη∈C the integral αdηis well defined.We equip H with the coarsest topology such that the function(α,η)∈H×C→ αdηis continuous. This then corresponds to the topology of uniform convergence for continuous functions f on T1M if we represent a current by aΦt-invariantfinite Borel measure on T1M and a H¨o lder class by a H¨o lder continuous function on T1M.Consider again the geodesicflowΦt on the unit tangent bundle T1M of M.Let M be the compact convex space ofΦt-invariant Borel-probability measures on T1M equippedwith the weak*-topology.Recall that the pressure pr(f)of a continuous function f on T1M is defined by pr(f)=sup{hν− fdν|ν∈M}where hνis the entropy ofν.Call a H¨o lder class normalized if it can be represented by a function f with pr(f)=0.Every normalized H¨o lder class is positive.Vice versa,ifαis a positive H¨o lder class,then there is a unique constant h(α)>0such that h(α)αis normalized.We call h(α)the topological entropy ofα.Identify the space offlip invariant normalized H¨o lder classes with the projectivization PH of H.Every element from PH determines its Gibbs current whose class in the space PC= C/R+of projective currents where R+acts on C via multiplication with a positive constant does not depend on any choices made.Thus we obtain a natural map of the space PH of projectiveflip invariant positive H¨o lder classes into the space PC of projective currents. We call the image of PH under this map the space of projective Gibbs currents.Also we call a current a Gibbs current if it corresponds to the Gibbs equilibrium state of a normalizedflip invariant H¨o lder class on T1M.Denote by[µα]the class in PC defined by the Gibbs state of h(α)αwhereα∈H and as before,h(α)is the topological entropy ofα.Clearly h(aα)=a−1h(α)for all a>0,moreover ifαis the length cocycle of a negatively curved Riemannian metric g on M then h(α)is just the topological entropy of the geodesicflow of g.First of all we have:Lemma2.1:h(α+β)≤h(α)h(β)/(h(α)+h(β))with equality if and only ifαis a constant multiple ofβ.Proof:Fix a negatively curved metric g on M with geodesicflowΦt.Letα,βbe normalized H¨o lder classes.Letµbe the uniqueΦt-invariant Borel probability measure on the unit tangent bundle for(M,g)which is a Gibbs equilibrium state for the normalization ofα+βand denote by hµthe entropy ofµ.Then we have αdµ≥hµ/h(α), βdµ≥hµ/h(β)with equality if and only ifα=β.Then [α+β]dµ≥hµ(1/h(α)+1/h(β))and hence the lemma follows from the fact that hµ=h(α+β) (α+β)dµ.q.e.d.We equip PC with the topology induced from the weak*-topology on C.Lemma2.2:i)The topological entropy h:H→(0,∞)is continuous.ii)The map [α]∈PH→[µα]∈PC is continuous.Proof:Let{αi}⊂H be a sequence converging to someα∈H.For i≥1letηi∈M be the unique Gibbs equilibrium state for h(αi)αi.Since M is compact we may assume by passing to a subsequence that the measuresηi converge weakly to someη∈M.If hνdenotes again the entropy ofν∈M thenν→hνis upper semi-continuous and hence .hη≥lim i→∞sup hηiBy the definition of the topology on H we haveαi dηi→ αdη>0(i→∞),in particular {h (αi )}is bounded from above and below by a positive constant.By passing to a subsequence we may assume that h (αi )→¯h >0(i →∞).Then h η−¯h αdη≥lim sup i →∞h ηi −h (αi ) αi dηi =0and consequently ¯h ≤h (α).On the other hand,if ¯h<h (α)then there is ν∈M with h ν−¯h αdν>0.Then also h ν−h (αi ) αi dν>0for i sufficiently large which is impossible.In other words,the function h is indeed continuous.Moreover,if η∈M is as above,then h η−¯hαdη=0and hence the current determined by ηis contained in the class of [µα].From this the continuity of the map [α]→[µα]is immediate.q.e.d.Recall from [H5]that a cross-ratio for Γ=π1(M )is a H¨o lder continuous positive function Cr on the space of quadruples of pairwise distinct points in ∂˜Mwith the following properties:1)Cr is invariant under the action of Γon (∂˜M)4.2)Cr (a,a ′,b,b ′)=Cr (a ′,a,b,b ′)−13)Cr (a,a ′,b,b ′)=Cr (b,b ′,a,a ′)4)Cr (a,a ′,b,b ′)Cr (a ′,a ′′,b,b ′)=Cr (a,a ′′,b,b ′)5)Cr (a,a ′,b,b ′)Cr (a ′,b,a,b ′)Cr (b,a,a ′,b ′)=1.Property 5)above is a consequence of properties 1)-4)and the fact that Cr admits a H¨o lder continuous extension to the space of quadruples (a,a ′,b,b ′)of points in ∂˜Mwhich satisfy {a,a ′}∩{b,b ′}=∅.This extension equals 1for every quadruple (a,a ′,b,b ′)for which either a =a ′or b =b ′(this was communicated to me by F.Ledrappier).We showed in [H5]that there is natural bijection between the space of cohomology classes of flip invariant H¨o lder cocyles and the space of cross ratios on ∂˜M×∂˜M −∆.We call a cross ratio Cr positive if it corresponds under this identification to a positive H¨o lder class.Lemma 2.3:Let Cr be a cross ratio on ∂˜M.Then Cr (a,b,c,d )−1+Cr (b,c,d,a )−1→1as a →b ,locally uniformly in (c,d )if and only if Cr is positive.Proof:Recall from [H5]that there is a H¨o lder continuous symmetric function (,)on ∂˜M ×∂˜M −∆(where ∆denotes the diagonal)such that log Cr (a,b,c,d )=(a,d )+(b,c )−(a,c )−(b,d )for pairwise distinct points a,b,c,d in ∂˜M .If Cr is positive then (a,b )→∞as a →b in ∂˜M.Now Cr (a,b,c,d )−1+Cr (b,c,d,a )−1=e (a,c )+(b,d )[e −(a,d )−(b,c )+e −(b,a )−(c,d )]and this converges to 1as a →b if and only if e −(b,a )−(c,d )→0as a →b .The formula also shows that this convergence is locally uniform in (c,d )if Cr is positive.q.e.d.From now on we specialize to the case that M is an oriented surface.The ideal boundary of its universal covering is naturally homeomorphic to S 1.Fix once and for all an orientation for the circle S 1.This orientation then determines for every ordered pair (a,b )of points a =b in S 1a unique half-open interval [a,b [⊂S 1with endpoints a and b .Call a quadruple (a 1,a 2,a 3,a 4)of pairwise distinct points in S 1ordered if a i is contained in [a i −1,a i +1[for i =1, (4)Proposition2.4:For every Gibbs currentµthere is a unique cross ratio Cr for Γ=π1(M)such that log Cr(a,b,c,d)=µ[a,b[×[c,d[for every ordered quadruple(a,b,c,d) in S1.Moreover Cr is positive.Proof:Let(a,b,c,d)be an ordered quadruple of pairwise distinct points in S1and define[a,b,c,d]=µ[a,b[×[c,d[.Sinceµis a current and hence invariant under exchange of the intervals[a,b[and[c,d[we have[c,d,a,b]=µ[c,d[×[a,b[=[a,b,c,d],i.e.Cr=e[] satisfies3)above.Moreover,if y∈]a,b[then(a,y,c,d)and(y,b,c,d)are ordered and (∗)µ[a,b[×[c,d[=µ[a,y[×[c,d[+µ[y,b[×[c,d[which corresponds to4)above for Cr on ordered quadruples.So far we have not used thatµis a Gibbs current;this now is essential to show that []is H¨o lder continuous.Fix a base point x∈˜M and view S1=∂˜M as the unit sphere in T x˜M.By formula(*)it is enough to show the following:For(c,d)∈S1×S1−∆and b∈]d,c[there are constantsβ>0,α>0such thatµ[a,b[×[c,d[≤β|a−b|αwhenever a is sufficiently close to b.For this choose a positive H¨o lder function f on T1M such thatµis the Gibbs current for f.Recall from[H5]that f induces a symmetric functionαf on S1×S1−∆such that β−1αf(a,b)1/χ≤(a|b)≤βαf(a,b)χwhere(|)is the Gromov product on S1×S1−∆with respect to the base-point x which defines the H¨o lder structure on S1=∂˜M and χ∈(0,1),β>0arefixed constants.Moreover for a sufficiently close to b,µ[a,b[×[c,d[is bounded from above by a constant multiple ofαf(a,b).From this H¨o lder continuity of[] on ordered quadruples of points in∂˜M is immediate.Next,if(a,b,c,d)is ordered,then we define[b,a,c,d]=−[a,b,c,d];this then implies also that[a,b,c,d]=−[a,b,d,c].Finally if we put[b,a,d,c]=[a,b,c,d]whenever(a,b,c,d) is ordered then we obtain an extension of[]to all quadruples of pairwise distinct points in∂˜M which is independent of the choice of an orientation for∂˜M.Moreover Cr=e[] clearly satisfies all defining properties of a cross ratio.q.e.d.We call the cross ratio Cr defined as above by a Gibbs currentµthe intersection cross ratio ofµand we write[]µ=log Cr.To justify this notion,recall that the intersection form i is a continuous bilinear form on the space of geodesic currents where the space of currents is equipped with the weak*-topology(see[B2]).Recall also that every free homotopy class[γ]in M defines a unique geodesic current which we denote again by[γ] as follows:Represent[γ]by a closed geodesic in M.The lifts of this geodesic to˜M define aΓ=π1(M)-invariant subset of the space∂˜M×∂˜M−∆of geodesics whose intersection with every compact subset of∂˜M×∂˜M−∆isfinite.Then[γ]is the sum of all Dirac masses on all points of∂˜M×∂˜M−∆corresponding to those lifts.Recall also that a free homotopy class in M is nothing else but a conjugacy class inπ1(M).Then we have:Lemma2.5:Letµbe a Gibbs current,letΨ∈Γand denote by a,b thefixed points of the action ofΨon S1.Then for everyξ∈]b,a[,the intersection ofµwith the conjugacyclass ofΨequals|[a,b,ξ,Ψξ]µ|.Proof:By definition,ifξis such that(a,b,ξ,Ψξ)is ordered,then[a,b,ξ,Ψξ]µ=µ[a,b[×[ξ,Ψξ[;but the right hand side of this equation just equals the intersection ofµwith current defined by the conjugacy class ofΨ(see[O]).q.e.d.Using Lemma2.5,we observe next that the intersection cross ratio just describes the duality between the Lebesgue-Liouville current of a metric g of negative curvature on M and the Bowen-Margulis current of the geodesicflow for g.For this we recall again that there is a natural1-1-correspondence betweenflip invariant H¨o lder classes and cross ratios ([H5]).Corollary 2.6:Letλg be the Lebesgue Liouville current of a metric g on M of negative curvature.Then the intersection cross ratio ofλg is twice the cross ratio defined by the length element of g.Proof:The computations in[B1]show that the intersection of the Lebesgue Liouville currentλg of g with a free homotopy classγin M is just twice the length of the closed geodesic with respect to g which representsγ.In other words,the value atγof the H¨o lder cohomology class corresponding to the intersection cross ratio ofλg equals twice the value atγof the length cocycle of the metric g.Since a H¨o lder class is determined by its values on free homotopy classes,the corollary follows.q.e.d.Another consequence of Proposition2.4is the following.Lemma2.7:The map[α]∈PH→[µα]∈PC is injective.Proof:Letα∈H be normalized and letµ∈[µα].LetΨ∈Γ=π1(M)and let a be the attractingfix point for the action ofΨon∂˜M,b be the repellingfix point.For fixedξ∈]b,a[we haveα(Ψ,a)=−lim k→∞1ofΓ,the measureηis in fact a geodesic current.We callηthe intersection current of Cr. In general however thefinitely additive function onfinite unions of products of right-half open intervals in S1is notσ-additive,which means that it is not a signed geodesic current (the formal difference of two geodesic currents).However we always callηthe intersection current of Cr.The results of[H5]show that every H¨o lder classαon T1M defines a unique intersection currentνα.The assignmentα→ναis linear,i.e.ifα,βare H¨o lder classes and if a,b∈R thenνaα+bβ=aνα+bνβ.Clearly if the intersection currentναis positive,i.e.ifναis in fact a current,thenαis a positive H¨o lder class.The reverse however is not true.To see this letα1,α2be the H¨o lder classes of the length cocyles of hyperbolic metrics g1=g2on M.Since the metrics g1,g2are bilipschitz equivalent there is a numberǫ>0such thatα1−ǫα2is a positive H¨o lder class.By Corollary2.6the intersection currentλi ofαi is1For this recallfirst of all that a free homotopy class in M which can be represented by a simple closed curve defines the projective class of a measured geodesic lamination. On the other hand,projective measured laminations form the boundary in PC of the projectivizations of the Lebesgue Liouville currents of hyperbolic metrics on M([B1]). Thus we may restrict our attention to free homotopy classes which can not be represented by simple closed curves.Fix a hyperbolic metric g on M and letγbe a closed geodesic in(M,g)with self intersections.By eventually changing the metric g we may assume thatγhas only double points.For simplicity we consider only the case thatγhas exactly one double point.The con-struction given below is also valid in the general case.Letγ:[0,T]→M be a parametriza-tion by arc length such thatγ(0)=γ(τ)=γ(T)for someτ∈(0,T)(in other words,γ(0) is the double point).Forǫ>0let Q(ǫ)be theǫ-neighborhood ofγin M.For sufficiently smallǫ,Q(ǫ)−γ[0,τ]has exactly two components,one of which,say Q1,is homeomorphic to an annulus. Choose a normal vectorfield t→N(t)alongγwhich points on(0,τ)inside Q1.Let m>0 be a large number;then there areǫ1(m),ǫ2(m)∈(0,ǫ)such that expǫ1(m)N(1/m)= expǫ1(m)N(τ−1/m)∈Q1(where exp is the exponential map of(M,g))andexp(−ǫ2(m)N(1/m))=expǫ2(m)N(τ+1/m)).Observe that the absolute values ofǫ1(m) andǫ2(m)only depend on m and the angle betweenγ′(0)and−γ′(τ)atγ(0).Thus for each sufficiently large m≥1we obtain a closed neighborhood A(m)ofγin M with the following properties:i)A(m)⊃A(m+1)and∩m≥1A(m)=γii)A(m)=∪4i=0A i(m)where the sets A i(m)are closed with piecewise smooth boundary and pairwise disjoint interior.iii)A0(m)is a geodesic quadrangle in M containingγ(0)in its interior,with verticesexpǫ1(m)N(1/m)=x1(m),exp−ǫ2(m)N(1/m)=x2(m), exp(−ǫ1(m)N(T−1/m))=x3(m)and expǫ2(m)N(T−1/m)=x4(m).iv)The boundary of A i(m)(i=1,...,4)contains two smooth geodesic segments of length ǫ1(m)orǫ2(m)which meet at x i(m)and are subarcs of the boundary of A0(m).It also contains a subarc ofγ.With respect to normal exponential coordinates based at γthe metric on A i(m)can be written in the form(cosh s)2dt2+ds2(s∈[0,ǫ1(m)]or s∈[0,ǫ2(m)]).Change now the metric in the interior of A(m)as follows:Fix m≥1sufficiently large and for i=1,2choose a diffeomorphismΨi,m:[0,m]→[0,ǫi(m)]with the following properties:。