Constant Mean Curvature Hypersurfaces in the (n+1)-Sphere by Gluing Spherical Building Bloc

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Constant scalar curvature Kahler metrics on fibred complex surfaces

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JOEL FINE
The surfaces covered by Theorem 1.1 are each a family of stable curves parameterised by a stable base, and so stand a good chance of being stable, at least for a polarisation in which the base is large. This is certainly true in the analogous situation for bundles. The algebro-geometric view point is not explored here, except for a few comments relating analytic observations to stability conditions. 1.2. Projectivised vector bundles. A result analogous to Theorem 1.1, but concerning projectivised vector bundles, is proved by Hong in [6] and [7]. The techniques used there are similar to those applied here. There are two important diﬀerences between the situations, however. In both cases the ﬁbres have cscK metrics. However, in Theorem 1.1 the ﬁbres have no nonzero holomorphic vector ﬁelds, whilst the ﬁbres of a projectivised vector bundle have holomorphic vector ﬁelds which are not Killing. If (F, ω ) is a compact K¨ ahler manifold with cscK metric, and ξ is a holomorphic vector ﬁeld on F which is not Killing, then ﬂowing ω along ξ gives a non-trivial family of cohomologous cscK metrics. From the point of view of the analysis, this situation leads to problems, as is explained later. It is avoided in [6] and [7] by the additional assumption that, whilst there are holomorphic vector ﬁelds on the ﬁbres, they are not induced by a global holomorphic vector ﬁeld. From the point of view of the algebraic geometry, this is related to the stability of the vector bundle which is projectivised. The other important diﬀerence between the two situations is that the ﬁbres of a ruled manifold are rigid, whilst the ﬁbres in Theorem 1.1 have moduli. This leads to extra considerations in the proof, which are not required in Hong’s work. 1.3. Base genus 0 or 1. Let X satisfy the hypotheses of Theorem 1.1. When Σ has genus 0 or 1, a direct proof of the theorem can be given. Theorem 1.2. Let X be a compact connected complex surface admitting a holomorphic submersion π : X → Σ to a surface of genus 0 or 1, for which all the ﬁbres are of genus at least 2. Then X admits cscK metrics. Proof. Mapping each ﬁbre of π to its Jacobian determines a map j : Σ → A , where A denotes the moduli space of principally polarised abelian varieties (of dimension equal to the genus of the ﬁbres of π ). The universal cover of A is the Siegel upper half space S which can be realised as a bounded domain in CN . The map j lifts to a holomorphic map from the universal cover of Σ to S . Since g(Σ) = 0 or 1, this map must be constant: when g(Σ) = 0 the map is a holomorphic map from a compact manifold; when g(Σ) = 1 the map is a bounded holomorphic map from C. Hence j is constant. By Torelli’s theorem, all the ﬁbres of π are biholomorphic. As the model ﬁbre S of X → Σ has genus at least 2, its group of biholomorphisms Γ is ﬁnite. Deﬁne a principal Γ bundle P → Σ by setting the

Reality problems in the soliton theory

REALITY PROBLEMS IN THE SOLITON THEopological Charge can be naturally extended to all real generic regular ﬁnite-gap (quasiperiodic) solutions. It is the most basic conservation law. Problem: How to calculate topological charge of the real ﬁnite-gap solutions in terms of the spectral data. Let us remind, that the Inverse Scattering (Spectral) Data for KdV and Sine-Gordon systems consist of a Riemann surface (Spectral Curve) Γ with ﬁnite genus equals to g and a collection of points (“divisor”) D = γ1 + . . . + γg . (For NLS and some other systems number of poles may be diﬀerent from genus). In the case of KdV (or ﬁnite-gap periodic Schr¨ odinger operator L = 2 −∂x + u(x)) this surface Γ is hyperelliptic. In the case of the SineGordon equation the surface is also hyperelliptic, µ2 = λ

黎曼空间型中具有常数量曲率的超曲面的刚性

黎曼空间型中具有常数量曲率的超曲面的刚性刘建成;谢逊【摘要】The compact hypersurfaces with constant scalar curvature in a Riemannian space form are studied , and an estimate of constant scalar curvature is obtained . As a result of this estimation , a rigidity theorem of such hypersurfaces is proved .%设 Mn为等距浸入到黎曼空间型N n＋1（c）中的具有常数量曲率的紧致超曲面，得到了数量曲率的一个估计，并应用它证明了该类超曲面的一个刚性分类结果。

【期刊名称】《西北师范大学学报（自然科学版）》【年(卷),期】2015(000)006【总页数】4页(P17-20)【关键词】黎曼空间型;数量曲率;紧致;超曲面【作者】刘建成;谢逊【作者单位】西北师范大学数学与统计学院，甘肃兰州 730070;西北师范大学数学与统计学院，甘肃兰州 730070【正文语种】中文【中图分类】O186.12E-mail:**************.cnKey words:Riemannian space form;scalar curvature;compact;hypersurface 设Nn+1(c)是(n+1)维常截曲率为c的黎曼空间型.根据c>0,c=0或c<0,分别称之为球空间Sn+1(c),欧氏空间Rn+1或双曲空间Hn+1(c).欧氏空间中具有常数量曲率的紧致超曲面的研究始于Cheng-Yau[1].一个熟知的问题是:Rn+1中具有常数量曲率的紧致超曲面是否为球面?当n=2时该问题即为经典的Liebmann定理；而当n>2时,Cheng-Yau证明了若超曲面M的截曲率K(M)>0,则结论正确；1991年，Montiel-Ros[2]证得:对于欧氏空间中的紧致连通嵌入超曲面,若有一个r阶平均曲率Hr等于常数,则Mn为全脐的,从而是一个超球面.因此在欧氏空间中,具有常数量曲率的紧致超曲面的截曲率K(M)>0是必然的. 上述问题在黎曼空间型中的推广即为本文的研究方向,在这方面已有诸多刚性结果.文献[3]研究标准球面中具有两个不同主曲率,且r阶平均曲率为常数的超曲面.若这两个不同主曲率的重数均大于1,则该超曲面等距于Sk(a)×Sn-k(b),2≤k≤n-1.文献[4]根据Cheng-Yau引入的自伴微分算子获得了具有常数量曲率的完备超曲面中,若其截曲率非负,则超曲面或者为全脐的,或者为黎曼乘积)×Sn-k(r),2≤k≤n-1.文献[5,6]分别获得了在常截曲率为c的黎曼空间型中,若c≥0且Mn的Ricci曲率不小于(n-1)c,或者c<0且Mn的截曲率为正,则Mn为全脐点超球面.目前,对于Sn+1(c)中具有常数量曲率超曲面Mn的分类问题还尚未解决,仅在Mn为旋转超曲面时,得到了一些结果[7].近年来,许多学者一直致力于曲率或第二基本形式的拼挤研究[8-10].本文首先证明在黎曼空间型Nn+1(c)中具有常数量曲率的紧致超曲面的数量曲率r≥n(n-1)c，然后利用这个结论给出该类超曲面的一个刚性分类.定理1 设Mn是Nn+1(c)中数量曲率r为常数的紧致定向超曲面,则必有r-n(n-1)c≥0,当且仅当Mn为全测地时等号成立.定理2 设Mn为Nn+1(c)的紧致超曲面,S为第二基本形式模长的平方.若Mn具有常数量曲率,则当c≥0时Mn为全脐的;当c<0时,若,则Mn或者为全脐的,或者等距于双曲柱面H1(1-coth2t)×Sn-1(1-tanh2t),其中t为任意正常数.设为超曲面Mn到黎曼空间型Nn+1(c)中的等距浸入,且Mn紧致定向.在Nn+1(c)中选取局部单位正交标架场{e1,…,en,en+1},使得限制在Mn上{e1,…,en}与Mn相切,en+1与Mn正交.取{ω1,…,ωn,ωn+1}为{e1,…,en,en+1}的对偶标架场,约定指标取值范围1≤i,j,…≤n,则Mn的Darboux标架运动方程为根据Cartan引理有,其中hij=hji.记Rijkl为Mn的曲率张量的分量,则由Gauss方程有Rijkl=c(δikδjl-δilδjk)+(hikhjl-hilhjk). (1)若⊗ωj为Mn的第二基本形式,则为Mn的平均曲率,为Mn的第二基本形式模长平方.由(1)式可知进而,用hijk,hijkl分别表示hij的一阶和二阶共变导数的分量,则有于是有Codazzi方程hijk=hikj和Ricci恒等式定义hij的Laplacian为,则记平均曲率H的一阶、二阶共变导数分别为.对于Mn上的C2函数f,定义其梯度、Hessian算子及方框算子□[1]如下:取φij=hij-Hδij,则φφijwi⊗wj为对称张量.记φ,则φ.显然φ非负,并且φ当且仅当Mn为全脐超曲面.引理1[11] 设Mn为Nn+1(c)的超曲面,则引理2[12] 设Mn是Nn+1(c)中具有常数量曲率的超曲面.若r-n(n-1)c≥0,则引理3[13] 设μi(i=1,…,n)为n个实数,使得且 2,其中β≥0为常数，则有等号成立当且仅当至少有n-1个μi相等.定理1的证明已知⊂Rn+2为等距浸入.令x,x,x∈Mn,则f是Mn上的光滑函数.由Mn的紧致性,必有一点x0∈Mn,使得f(x)在x0处达到最大值,所以f的一阶、二阶绝对微分分别满足因为所以由(5),(6)式可知即x0正好是曲面Mn在点x0处的法矢量.微分x,ei,并根据运动方程得根据(7)式有于是fij=hijx,en+1,从而由(5)式有由于x0是超曲面Mn在x0处的法矢量,故于是,由(8)式知Mn在x0处的第二基本形式是恒定的二次型.选取适当的标架,使得hij=λiδij.于是,当x,en+1x0<0时,由(8),(9)式知,第二基本形式h恒半正定.根据对称矩阵半正定的充要条件是它的特征值全为正数或零可知,Mn的所有主曲率λi(i=1,2,…,n)均为非负数.同理,当x,en+1x0>0时,第二基本形式h恒半负定,即Mn的所有主曲率λi(i=1,2,…,n)均为非正数.两种情形都表明Mn的主曲率定号,从而其数量曲率与Nn+1(c)数量曲率之差即为当且仅当Mn为全测地时,等号成立.由于Mn的数量曲率为常数,从而在整体上r-n(n-1)c为非负常数. 】推论1 设Mn为黎曼空间型Nn+1(c)中具有常数量曲率r的紧致超曲面,则数量曲率r满足左侧等号成立当且仅当Mn为全测地的,右侧等号成立当且仅当Mn为全脐的. 可以看出H=0时,空间型中具有常数量曲率的超曲面为全测地的.定理2的证明由(4)式知而根据Weitzenböck公式有2+n2HΔH,所以另一方面,从(2)式可得Δ(n2H2)=ΔS,所以由引理1及(10)式可得利用引理2,引理3,从上式得到( i )当c>0时,根据(2)式及定理 1,有结合(11)式便有□φ.由于上式右边非负,且∫Mn□(nH)dV=0,所以φ,等价于Mn为全脐超曲面,从而Mn为一个超球面.再根据Gauss方程知,Mn具有常截曲率(c+H2)当且仅当H=0时Mn为全测地的.( ii )当c<0时,仿照文献[14]的方法,考虑以为特征值的二次型作正交变换则由于变换为正交变换,从而x2+y2=u2+v2.令φ,于是x2+y2=S,且再由(11)式知□φ若,则由(12)式知φ,即S=nH2,从而Mn为全脐的.若,代入Gauss方程(2)知,平均曲率H为常数,从而有于是hijk=0,Mn具有平行第二基本形式,且主曲率λi为常数.由引理3知,Mn是至多有两个不同常主曲率的等参超曲面,且其中一个重数为1.当λ1=λ2=…=λn=H时,情形同上,Mn为全脐超曲面.当Mn有两个不同主曲率时,由文献[15]知,Mn等距于双曲柱面H1(1-coth2t)×Sn-1(1-tanh2t),其中t为正常数.(iii)当c=0时,由文献[2]知Mn为全脐的.综合以上,定理2得证. 】【相关文献】[1] CHERN Shiing-shen,YAU Shing-tung.Hypersurfaces with constant scalarcurvature[J].Trans Amer Math Soc,1977,225(3):195.[2] MONTIEL S,ROS pact hypersurfaces:The Alexandrov theorem for higher order mean curvatures[J].J Diff Geom,1991,52:279.[3] CHENG Qing-ming,LI Hai-zhong,WEI Guo-xin.Embedded hypersurfaces with constant m-th mean curvature in a unit sphere[J].Commun Contemp Math,2010,12(6):997.[4] BRASIL Jr A,COLARES A G,PALMAS plete hypersurfaces with constant scalar curvature in spheres[J].Monatsh Math,2010,161(4):369.[5] ROS pact hypersurfaces with constant scalar curvature and a congruence theorem[J].J Diff Geom,1988,27(2):215.[6] YAU Shing-tung.Submanifolds with constant mean curvature(Ⅱ)[J].Amer JMath,1975,97(1):76.[7] MARIA L L.Rotational hypersurfaces of space forms with constant scalarcurvature[J].Manus Math,1990,67(1):285.[8] XU Yi-wen,XU Zhi-yuan.On rigidity of Clifford torus in a unit sphere[J].Appl Math J Chinese Univ,2011,26(1):121.[9] LIU Xi-min,SU Wei-hong.Hypersurfaces with constant scalar curvature in a hyperbolic space forms[J].Balkan J Geom Appl,2002,7(1):121.[10] WEI Si-ming,XU Hong-wei.Scalar curvature of minimal hypersurfaces in asphere[J].Math Res Lett,2007,14(3):423.[11] CHENG Qing-ming,ISHIKAWA S.Spacelike hypersurfaces with constant scalar curvature[J].Manus Math,1998,95(4):499.[12] LI Hai-zhong.Global rigidity theorems of hypersurfaces[J].Ark Mat,1997,35(4):327.[13] OKUMURA K.Hypersurfaces and a pinching problem on the second fundamental tensor[J].Amer J Math,1974,96(1):207.[14] HOU Zhong-hua.Hypersurfaces in a sphere with constant mean curvature[J].Proc Amer Math Soc,1997,125(4):1193.[15] MONTIEL S.An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature[J].Indiana Univ MathJ,1988,37(4):909.。

分析化学专业英语词汇总结

专业英语词汇-----分析化学第一章绪论分析化学：analytical chemistry定性分析：qualitative analysis定量分析：quantitative analysis物理分析：physical analysis物理化学分析：physico-chemical analysis仪器分析法：instrumental analysis流动注射分析法：flow injection analysis；FIA顺序注射分析法：sequentical injection analysis;SIA化学计量学：chemometrics第二章误差的分析数据处理绝对误差：absolute error相对误差：relative error系统误差：systematic error可定误差：determinate error随机误差：accidental error不可定误差：indeterminate error准确度：accuracy精确度：precision偏差：debiation，d平均偏差：average debiation相对平均偏差：relative average debiation标准偏差（标准差）：standerd deviation;S相对平均偏差：relatibe standard deviation;RSD变异系数：coefficient of variation误差传递：propagation of error有效数字：significant figure置信水平：confidence level显著性水平：level of significance合并标准偏差（组合标准差）：pooled standard debiation 舍弃商：rejection quotient ;Q化学定量分析第三章滴定分析概论滴定分析法：titrametric analysis滴定：titration容量分析法:volumetric analysis化学计量点：stoichiometric point等当点：equivalent point电荷平衡：charge balance电荷平衡式：charge balance equation质量平衡：mass balance物料平衡：material balance质量平衡式：mass balance equation第四章酸碱滴定法酸碱滴定法：acid-base titrations 质子自递反应：auto protolysis reaction质子自递常数：autoprotolysis constant质子条件式:proton balance equation酸碱指示剂：acid-base indicator指示剂常数：indicator constant变色范围：colour change interval混合指示剂：mixed indicator双指示剂滴定法：double indicator titration第五章非水滴定法非水滴定法：nonaqueous titrations质子溶剂：protonic solvent酸性溶剂：acid solvent碱性溶剂：basic solvent两性溶剂:amphototeric solvent无质子溶剂：aprotic solvent均化效应：differentiatin g effect区分性溶剂：differentiating solvent离子化：ionization离解：dissociation结晶紫：crystal violet萘酚苯甲醇: α-naphthalphenol benzyl alcohol奎哪啶红：quinadinered百里酚蓝：thymol blue偶氮紫：azo violet溴酚蓝：bromophenol blue第六章配位滴定法配位滴定法：compleximetry乙二胺四乙酸：ethylenediamine tetraacetic acid,EDTA 螯合物：chelate compound金属指示剂：metal lochrome indcator第七章氧化还原滴定法氧化还原滴定法：oxidation-reduction titration碘量法：iodimetry溴量法：bromimetry ]溴量法：bromine method铈量法：cerimetry高锰酸钾法：potassium permanganate method条件电位：conditional potential溴酸钾法：potassium bromate method硫酸铈法：cerium sulphate method偏高碘酸：metaperiodic acid高碘酸盐：periodate亚硝酸钠法：sodium nitrite method重氮化反应：diazotization reaction重氮化滴定法：diazotization titration亚硝基化反应：nitrozation reaction亚硝基化滴定法：nitrozation titration外指示剂：external indicator外指示剂：outside indicator重铬酸钾法：potassium dichromate method 第八章沉淀滴定法沉淀滴定法：precipitation titration容量滴定法：volumetric precipitation method 银量法：argentometric method第九章重量分析法重量分析法：gravimetric analysis挥发法：volatilization method引湿水（湿存水）：water of hydroscopicity 包埋(藏)水：occluded water吸入水：water of imbibition结晶水：water of crystallization组成水：water of composition液-液萃取法：liquid-liquid extration溶剂萃取法：solvent extration反萃取：counter extraction分配系数：partition coefficient分配比：distribution ratio离子对（离子缔合物）：ion pair沉淀形式：precipitation forms称量形式：weighing forms仪器分析概述物理分析：physical analysis物理化学分析：physicochemical analysis仪器分析：instrumental analysis第十章电位法及永停滴定法电化学分析：electrochemical analysis电解法：electrolytic analysis method电重量法：electrogravimetry库仑法：coulo metry库仑滴定法：coulo metric titration电导法：conductometry电导分析法：conductometric analysis电导滴定法：conductometric titration电位法：potentiometry直接电位法：dirext potentiometry电位滴定法：potentiometric titration伏安法：voltammetry极谱法：polarography溶出法：stripping method电流滴定法：amperometric titration化学双电层：chemical double layer相界电位：phase boundary potential 金属电极电位：electrode potential化学电池：chemical cell液接界面：liquid junction boundary原电池：galvanic cell电解池：electrolytic cell负极：cathode正极：anode电池电动势：eletromotive force指示电极：indicator electrode参比电极：reference electroade标准氢电极：standard hydrogen electrode一级参比电极：primary reference electrode饱和甘汞电极：saturated calomel electrode银－氯化银电极：silver silver-chloride electrode液接界面：liquid junction boundary不对称电位：asymmetry potential表观PH值：apparent PH复合PH电极：combination PH electrode离子选择电极：ion selective electrode敏感器：sensor晶体电极:crystalline electrodes均相膜电极：homogeneous membrance electrodes非均相膜电极：heterogeneous membrance electrodes非晶体电极：non- crystalline electrodes刚性基质电极：rigid matrix electrode流流体载动电极：electrode with a mobile carrier气敏电极：gas sensing electrodes酶电极：enzyme electrodes金属氧化物半导体场效应晶体管：MOSFET离子选择场效应管：ISFET总离子强度调节缓冲剂：total ion strength adjustment buffer,TISAB永停滴定法：dead-stop titration双电流滴定法（双安培滴定法）：double amperometric titration 第十一章光谱分析法概论普朗克常数：Plank constant电磁波谱：electromagnetic spectrum光谱：spectrum光谱分析法：spectroscopic analysis原子发射光谱法：atomic emission spectroscopy质量谱：mass spectrum质谱法：mass spectroscopy,MS第十二章紫外－可见分光光度法紫外－可见分光光度法：ultraviolet and visible spectrophotometry;UV-vis肩峰：shoulder peak末端吸收：end absorbtion生色团：chromophore助色团：auxochrome红移：red shift长移：bathochromic shift短移：hypsochromic shift蓝（紫）移：blue shift增色效应（浓色效应）：hyperchromic effect减色效应（淡色效应）：hypochromic effect强带：strong band弱带：weak band吸收带：absorption band透光率：transmitance,T吸光度：absorbance谱带宽度：band width杂散光：stray light噪声：noise暗噪声：dark noise散粒噪声：signal shot noise闪耀光栅：blazed grating全息光栅：holographic grating光二极管阵列检测器：photodiode array detector 偏最小二乘法：partial least squares method ,PLS褶合光谱法：convolution spectrometry褶合变换：convolution transform,CT离散小波变换：wavelet transform,WT多尺度细化分析：multiscale analysis供电子取代基：electron donating group吸电子取代基：electron with-drawing group第十三章荧光分析法荧光：fluorescence荧光分析法：fluorometryX-射线荧光分析法：X-ray fluorometry原子荧光分析法：atomic fluorometry分子荧光分析法：molecular fluorometry振动弛豫：vibrational relaxation内转换：internal conversion外转换：external conversion体系间跨越：intersystem crossing激发光谱：excitation spectrum荧光光谱：fluorescence spectrum斯托克斯位移：Stokes shift荧光寿命：fluorescence life time荧光效率：fluorescence efficiency荧光量子产率：fluorescence quantum yield荧光熄灭法：fluorescence quenching method散射光：scattering light瑞利光：R a yleith scattering light拉曼光：Raman scattering lightAbbe refractometer 阿贝折射仪absorbance 吸收度absorbance ratio 吸收度比值absorption 吸收absorption curve 吸收曲线absorption spectrum 吸收光谱absorptivity 吸收系数accuracy 准确度acid-dye colorimetry 酸性染料比色法acidimetry 酸量法acid-insoluble ash 酸不溶性灰分acidity 酸度activity 活度第十四章色谱法additive 添加剂additivity 加和性adjusted retention time 调整保留时间adsorbent 吸附剂adsorption 吸附affinity chromatography 亲和色谱法aliquot （一）份alkalinity 碱度alumina 氧化铝ambient temperature 室温ammonium thiocyanate 硫氰酸铵analytical quality control（AQC）分析质量控制anhydrous substance 干燥品anionic surfactant titration 阴离子表面活性剂滴定法antibiotics-microbial test 抗生素微生物检定法antioxidant 抗氧剂appendix 附录application of sample 点样area normalization method 面积归一化法argentimetry 银量法arsenic 砷arsenic stain 砷斑ascending development 上行展开ash-free filter paper 无灰滤纸（定量滤纸）assay 含量测定assay tolerance 含量限度atmospheric pressure ionization(API) 大气压离子化attenuation 衰减back extraction 反萃取back titration 回滴法bacterial endotoxins test 细菌内毒素检查法band absorption 谱带吸收baseline correction 基线校正baseline drift 基线漂移batch, lot 批batch(lot) number 批号Benttendorff method 白田道夫（检砷）法between day (day to day, inter-day) precision 日间精密度between run (inter-run) precision 批间精密度biotransformation 生物转化bioavailability test 生物利用度试验bioequivalence test 生物等效试验biopharmaceutical analysis 体内药物分析，生物药物分析blank test 空白试验boiling range 沸程British Pharmacopeia (BP) 英国药典bromate titration 溴酸盐滴定法bromimetry 溴量法bromocresol green 溴甲酚绿bromocresol purple 溴甲酚紫bromophenol blue 溴酚蓝bromothymol blue 溴麝香草酚蓝bulk drug, pharmaceutical product 原料药buret 滴定管by-product 副产物calibration curve 校正曲线calomel electrode 甘汞电极calorimetry 量热分析capacity factor 容量因子capillary zone electrophoresis (CZE) 毛细管区带电泳capillary gas chromatography 毛细管气相色谱法carrier gas 载气cation-exchange resin 阳离子交换树脂ceri(o)metry 铈量法characteristics, description 性状check valve 单向阀chemical shift 化学位移chelate compound 鳌合物chemically bonded phase 化学键合相chemical equivalent 化学当量Chinese Pharmacopeia (ChP) 中国药典Chinese material medicine 中成药Chinese materia medica 中药学Chinese materia medica preparation 中药制剂Chinese Pharmaceutical Association (CPA) 中国药学会chiral 手性的chiral stationary phase (CSP) 手性固定相chiral separation 手性分离chirality 手性chiral carbon atom 手性碳原子chromatogram 色谱图chromatography 色谱法chromatographic column 色谱柱chromatographic condition 色谱条件chromatographic data processor 色谱数据处理机chromatographic work station 色谱工作站clarity 澄清度clathrate, inclusion compound 包合物clearance 清除率clinical pharmacy 临床药学coefficient of distribution 分配系数coefficient of variation 变异系数color change interval （指示剂）变色范围color reaction 显色反应colorimetric analysis 比色分析colorimetry 比色法column capacity 柱容量column dead volume 柱死体积column efficiency 柱效column interstitial volume 柱隙体积column outlet pressure 柱出口压column temperature 柱温column pressure 柱压column volume 柱体积column overload 柱超载column switching 柱切换committee of drug evaluation 药品审评委员会comparative test 比较试验completeness of solution 溶液的澄清度compound medicines 复方药computer-aided pharmaceutical analysis 计算机辅助药物分析concentration-time curve 浓度－时间曲线confidence interval 置信区间confidence level 置信水平confidence limit 置信限congealing point 凝点congo red 刚果红（指示剂）content uniformity 装量差异controlled trial 对照试验correlation coefficient 相关系数contrast test 对照试验counter ion 反离子（平衡离子）cresol red 甲酚红（指示剂）crucible 坩埚crude drug 生药crystal violet 结晶紫（指示剂）cuvette, cell 比色池cyanide 氰化物cyclodextrin 环糊精cylinder, graduate cylinder, measuring cylinder 量筒cylinder-plate assay 管碟测定法daughter ion （质谱）子离子dead space 死体积dead-stop titration 永停滴定法dead time 死时间decolorization 脱色decomposition point 分解点deflection 偏差deflection point 拐点degassing 脱气deionized water 去离子水deliquescence 潮解depressor substances test 降压物质检查法derivative spectrophotometry 导数分光光度法derivatization 衍生化descending development 下行展开desiccant 干燥剂detection 检查detector 检测器developer, developing reagent 展开剂developing chamber 展开室deviation 偏差dextrose 右旋糖，葡萄糖diastereoisomer 非对映异构体diazotization 重氮化2,6-dichlorindophenol titration 2,6-二氯靛酚滴定法differential scanning calorimetry (DSC) 差示扫描热量法differential spectrophotometry 差示分光光度法differential thermal analysis (DTA) 差示热分析differentiating solvent 区分性溶剂diffusion 扩散digestion 消化diphastic titration 双相滴定disintegration test 崩解试验dispersion 分散度dissolubility 溶解度dissolution test 溶出度检查distilling range 馏程distribution chromatography 分配色谱distribution coefficient 分配系数dose 剂量drug control institutions 药检机构drug quality control 药品质量控制drug release 药物释放度drug standard 药品标准drying to constant weight 干燥至恒重dual wavelength spectrophotometry 双波长分光光度法duplicate test 重复试验effective constituent 有效成分effective plate number 有效板数efficiency of column 柱效electron capture detector 电子捕获检测器electron impact ionization 电子轰击离子化electrophoresis 电泳electrospray interface 电喷雾接口electromigration injection 电迁移进样elimination 消除eluate 洗脱液elution 洗脱emission spectrochemical analysis 发射光谱分析enantiomer 对映体end absorption 末端吸收end point correction 终点校正endogenous substances 内源性物质enzyme immunoassay(EIA) 酶免疫分析enzyme drug 酶类药物enzyme induction 酶诱导enzyme inhibition 酶抑制eosin sodium 曙红钠（指示剂）epimer 差向异构体equilibrium constant 平衡常数equivalence point 等当点error in volumetric analysis 容量分析误差excitation spectrum 激发光谱exclusion chromatography 排阻色谱法expiration date 失效期external standard method 外标法extract 提取物extraction gravimetry 提取重量法extraction titration 提取容量法extrapolated method 外插法，外推法factor 系数，因数，因子feature 特征Fehling’s reaction 费林反应field disorption ionization 场解吸离子化field ionization 场致离子化filter 过滤，滤光片filtration 过滤fineness of the particles 颗粒细度flame ionization detector(FID) 火焰离子化检测器flame emission spectrum 火焰发射光谱flask 烧瓶flow cell 流通池flow injection analysis 流动注射分析flow rate 流速fluorescamine 荧胺fluorescence immunoassay(FIA) 荧光免疫分析fluorescence polarization immunoassay(FPIA) 荧光偏振免疫分析fluorescent agent 荧光剂fluorescence spectrophotometry 荧光分光光度法fluorescence detection 荧光检测器fluorimetyr 荧光分析法foreign odor 异臭foreign pigment 有色杂质formulary 处方集fraction 馏分freezing test 结冻试验funnel 漏斗fused peaks, overlapped peaks 重叠峰fused silica 熔融石英gas chromatography(GC) 气相色谱法gas-liquid chromatography(GLC) 气液色谱法gas purifier 气体净化器gel filtration chromatography 凝胶过滤色谱法gel permeation chromatography 凝胶渗透色谱法general identification test 一般鉴别试验general notices （药典）凡例general requirements （药典）通则good clinical practices(GCP) 药品临床管理规范good laboratory practices(GLP) 药品实验室管理规范good manufacturing practices(GMP) 药品生产质量管理规范good supply practices(GSP) 药品供应管理规范gradient elution 梯度洗脱grating 光栅gravimetric method 重量法Gutzeit test 古蔡（检砷）法half peak width 半峰宽[halide] disk method, wafer method, pellet method 压片法head-space concentrating injector 顶空浓缩进样器heavy metal 重金属heat conductivity 热导率height equivalent to a theoretical plate 理论塔板高度height of an effective plate 有效塔板高度high-performance liquid chromatography (HPLC) 高效液相色谱法high-performance thin-layer chromatography (HPTLC) 高效薄层色谱法hydrate 水合物hydrolysis 水解hydrophilicity 亲水性hydrophobicity 疏水性hydroscopic 吸湿的hydroxyl value 羟值hyperchromic effect 浓色效应hypochromic effect 淡色效应identification 鉴别ignition to constant weight 灼烧至恒重immobile phase 固定相immunoassay 免疫测定impurity 杂质inactivation 失活index 索引indicator 指示剂indicator electrode 指示电极inhibitor 抑制剂injecting septum 进样隔膜胶垫injection valve 进样阀instrumental analysis 仪器分析insulin assay 胰岛素生物检定法integrator 积分仪intercept 截距interface 接口interference filter 干涉滤光片intermediate 中间体internal standard substance 内标物质international unit(IU) 国际单位in vitro 体外in vivo 体内iodide 碘化物iodoform reaction 碘仿反应iodometry 碘量法ion-exchange cellulose 离子交换纤维素ion pair chromatography 离子对色谱ion suppression 离子抑制ionic strength 离子强度ion-pairing agent 离子对试剂ionization 电离，离子化ionization region 离子化区irreversible indicator 不可逆指示剂irreversible potential 不可逆电位isoabsorptive point 等吸收点isocratic elution 等溶剂组成洗脱isoelectric point 等电点isoosmotic solution 等渗溶液isotherm 等温线Karl Fischer titration 卡尔·费歇尔滴定kinematic viscosity 运动黏度Kjeldahl method for nitrogen 凯氏定氮法Kober reagent 科伯试剂Kovats retention index 科瓦茨保留指数labelled amount 标示量leading peak 前延峰least square method 最小二乘法leveling effect 均化效应licensed pharmacist 执业药师limit control 限量控制limit of detection(LOD) 检测限limit of quantitation(LOQ) 定量限limit test （杂质）限度（或限量）试验limutus amebocyte lysate(LAL) 鲎试验linearity and range 线性及范围linearity scanning 线性扫描liquid chromatograph/mass spectrometer (LC/MS) 液质联用仪litmus paper 石蕊试纸loss on drying 干燥失重low pressure gradient pump 低压梯度泵luminescence 发光lyophilization 冷冻干燥main constituent 主成分make-up gas 尾吹气maltol reaction 麦牙酚试验Marquis test 马奎斯试验mass analyzer detector 质量分析检测器mass spectrometric analysis 质谱分析mass spectrum 质谱图mean deviation 平均偏差measuring flask, volumetric flask 量瓶measuring pipet(te) 刻度吸量管medicinal herb 草药melting point 熔点melting range 熔距metabolite 代谢物metastable ion 亚稳离子methyl orange 甲基橙methyl red 甲基红micellar chromatography 胶束色谱法micellar electrokinetic capillary chromatography(MECC, MEKC) 胶束电动毛细管色谱法micelle 胶束microanalysis 微量分析microcrystal 微晶microdialysis 微透析micropacked column 微型填充柱microsome 微粒体microsyringe 微量注射器migration time 迁移时间millipore filtration 微孔过滤minimum fill 最低装量mobile phase 流动相modifier 改性剂，调节剂molecular formula 分子式monitor 检测，监测monochromator 单色器monographs 正文mortar 研钵moving belt interface 传送带接口multidimensional detection 多维检测multiple linear regression 多元线性回归multivariate calibration 多元校正natural product 天然产物Nessler glasses(tube) 奈斯勒比色管Nessler’s r eagent 碱性碘化汞钾试液neutralization 中和nitrogen content 总氮量nonaqueous acid-base titration 非水酸碱滴定nonprescription drug, over the counter drugs (OTC drugs) 非处方药nonproprietary name, generic name 非专有名nonspecific impurity 一般杂质non-volatile matter 不挥发物normal phase 正相normalization 归一化法notice 凡例nujol mull method 石蜡糊法octadecylsilane chemically bonded silica 十八烷基硅烷键合硅胶octylsilane 辛（烷）基硅烷odorless 无臭official name 法定名official specifications 法定标准official test 法定试验on-column detector 柱上检测器on-column injection 柱头进样on-line degasser 在线脱气设备on the dried basis 按干燥品计opalescence 乳浊open tubular column 开管色谱柱optical activity 光学活性optical isomerism 旋光异构optical purity 光学纯度optimization function 优化函数organic volatile impurities 有机挥发性杂质orthogonal function spectrophotometry 正交函数分光光度法orthogonal test 正交试验orthophenanthroline 邻二氮菲outlier 可疑数据，逸出值overtones 倍频峰，泛频峰oxidation-reduction titration 氧化还原滴定oxygen flask combustion 氧瓶燃烧packed column 填充柱packing material 色谱柱填料palladium ion colorimetry 钯离子比色法parallel analysis 平行分析parent ion 母离子particulate matter 不溶性微粒partition coefficient 分配系数parts per million (ppm) 百万分之几pattern recognition 模式识别peak symmetry 峰不对称性peak valley 峰谷peak width at half height 半峰宽percent transmittance 透光百分率pH indicator absorbance ratio method? pH指示剂吸光度比值法pharmaceutical analysis 药物分析pharmacopeia 药典pharmacy 药学phenolphthalein 酚酞photodiode array detector(DAD) 光电二极管阵列检测器photometer 光度计pipeclay triangle 泥三角pipet(te) 吸移管，精密量取planar chromatography 平板色谱法plate storage rack 薄层板贮箱polarimeter 旋光计polarimetry 旋光测定法polarity 极性polyacrylamide gel 聚丙酰胺凝胶polydextran gel 葡聚糖凝胶polystyrene gel 聚苯乙烯凝胶polystyrene film 聚苯乙烯薄膜porous polymer beads 高分子多孔小球post-column derivatization 柱后衍生化potentiometer 电位计potentiometric titration 电位滴定法precipitation form 沉淀形式precision 精密度pre-column derivatization 柱前衍生化preparation 制剂prescription drug 处方药pretreatment 预处理primary standard 基准物质principal component analysis 主成分分析programmed temperature gas chromatography 程序升温气相色谱法prototype drug 原型药物provisions for new drug approval 新药审批办法purification 纯化purity 纯度pyrogen 热原pycnometric method 比重瓶法quality control(QC) 质量控制quality evaluation 质量评价quality standard 质量标准quantitative determination 定量测定quantitative analysis 定量分析quasi-molecular ion 准分子离子racemization 消旋化radioimmunoassay 放射免疫分析法random sampling 随机抽样rational use of drug 合理用药readily carbonizable substance 易炭化物reagent sprayer 试剂喷雾器recovery 回收率reference electrode 参比电极refractive index 折光指数related substance 有关物质relative density 相对密度relative intensity 相对强度repeatability 重复性replicate determination 平行测定reproducibility 重现性residual basic hydrolysis method 剩余碱水解法residual liquid junction potential 残余液接电位residual titration 剩余滴定residue on ignition 炽灼残渣resolution 分辨率，分离度response time 响应时间retention 保留reversed phase chromatography 反相色谱法reverse osmosis 反渗透rider peak 驼峰rinse 清洗，淋洗robustness 可靠性，稳定性routine analysis 常规分析round 修约（数字）ruggedness 耐用性safety 安全性Sakaguchi test 坂口试验salt bridge 盐桥salting out 盐析sample applicator 点样器sample application 点样sample on-line pretreatment 试样在线预处理sampling 取样saponification value 皂化值saturated calomel electrode(SCE) 饱和甘汞电极selectivity 选择性separatory funnel 分液漏斗shoulder peak 肩峰signal to noise ratio 信噪比significant difference 显著性差异significant figure 有效数字significant level 显著性水平significant testing 显著性检验silanophilic interaction 亲硅羟基作用silica gel 硅胶silver chloride electrode 氯化银电极similarity 相似性simultaneous equations method 解线性方程组法size exclusion chromatography(SEC) 空间排阻色谱法sodium dodecylsulfate, SDS 十二烷基硫酸钠sodium hexanesulfonate 己烷磺酸钠sodium taurocholate 牛璜胆酸钠sodium tetraphenylborate 四苯硼钠sodium thiosulphate 硫代硫酸钠solid-phase extraction 固相萃取solubility 溶解度solvent front 溶剂前沿solvophobic interaction 疏溶剂作用specific absorbance 吸收系数specification 规格specificity 专属性specific rotation 比旋度specific weight 比重spiked 加入标准的split injection 分流进样splitless injection 无分流进样spray reagent （平板色谱中的）显色剂spreader 铺板机stability 稳定性standard color solution 标准比色液standard deviation 标准差standardization 标定standard operating procedure(SOP) 标准操作规程standard substance 标准品stationary phase coating 固定相涂布starch indicator 淀粉指示剂statistical error 统计误差sterility test 无菌试验stirring bar 搅拌棒stock solution 储备液stoichiometric point 化学计量点storage 贮藏stray light 杂散光substituent 取代基substrate 底物sulfate 硫酸盐sulphated ash 硫酸盐灰分supercritical fluid chromatography(SFC) 超临界流体色谱法support 载体（担体）suspension 悬浊液swelling degree 膨胀度symmetry factor 对称因子syringe pump 注射泵systematic error 系统误差system model 系统模型system suitability 系统适用性tablet 片剂tailing factor 拖尾因子tailing peak 拖尾峰tailing-suppressing reagent 扫尾剂test of hypothesis 假设检验test solution(TS) 试液tetrazolium colorimetry 四氮唑比色法therapeutic drug monitoring(TDM) 治疗药物监测thermal analysis 热分析法thermal conductivity detector 热导检测器thermocouple detector 热电偶检测器thermogravimetric analysis(TGA) 热重分析法thermospray interface 热喷雾接口The United States Pharmacopoeia(USP) 美国药典The Pharmacopoeia of Japan(JP) 日本药局方thin layer chromatography(TLC) 薄层色谱法thiochrome reaction 硫色素反应three-dimensional chromatogram 三维色谱图thymol 百里酚（麝香草酚）（指示剂）thymolphthalein 百里酚酞（麝香草酚酞）（指示剂）thymolsulfonphthalein ( thymol blue) 百里酚蓝（麝香草酚蓝）（指示剂）titer, titre 滴定度time-resolved fluoroimmunoassay 时间分辨荧光免疫法titrant 滴定剂titration error 滴定误差titrimetric analysis 滴定分析法tolerance 容许限toluene distillation method 甲苯蒸馏法toluidine blue 甲苯胺蓝（指示剂）total ash 总灰分total quality control(TQC) 全面质量控制traditional drugs 传统药traditional Chinese medicine 中药transfer pipet 移液管turbidance 混浊turbidimetric assay 浊度测定法turbidimetry 比浊法turbidity 浊度ultracentrifugation 超速离心ultrasonic mixer 超生混合器ultraviolet irradiation 紫外线照射undue toxicity 异常毒性uniform design 均匀设计uniformity of dosage units 含量均匀度uniformity of volume 装量均匀性（装量差异）uniformity of weight 重量均匀性（片重差异）validity 可靠性variance 方差versus …对…，…与…的关系曲线viscosity 粘度volatile oil determination apparatus 挥发油测定器volatilization 挥发法volumetric analysis 容量分析volumetric solution(VS) 滴定液vortex mixer 涡旋混合器watch glass 表面皿wave length 波长wave number 波数weighing bottle 称量瓶weighing form 称量形式weights 砝码well-closed container 密闭容器xylene cyanol blue FF 二甲苯蓝FF（指示剂）xylenol orange 二甲酚橙（指示剂）zigzag scanning 锯齿扫描zone electrophoresis 区带电泳zwitterions 两性离子zymolysis 酶解作用簡體書目錄Chapter 1 Introduction 緒論1.1 The nature of analytical chemistry 分析化學的性質1.2 The role of analytical chemistry 分析化學的作用1.3 The classification of analytical chemistry分析化學的分類1.4 The total analytical process分析全過程Terms to understand重點內容概述Chapter 2 Errors and Data Treatment in Quantitative Analysis 定量分析中的誤差及數據處理2.1 Fundamental terms of errors誤差的基本術語2.2 Types of errors in experimental data實驗數據中的誤差類型2.2.1 Systematic errors 系統誤差2.2.2 Random errors偶然誤差2.3 Evaluation of analytical data分析數據的評價2.3.1 Tests of significance顯著性檢驗2.3.2 Rejecting data可疑值取捨2.4 Significant figures有效數字ProblemsTerms to understand重點內容概述Chapter 3 Titrimetric Analysis滴定分析法3.1 General principles基本原理3.1.1 Relevant terms of titrimetric analysis滴定分析相關術語3.1.2 The preparation of standard solution and the expression of concentration 標準溶液的配製與濃度表示方法3.1.3 The types of titrimetric reactions滴定反應類型3.2 Acid-base titration酸鹼滴定3.2.1 Acid-base equilibria 酸鹼平衡3.2.2 Titration curves滴定曲線3.2.3 Acid-base indicators酸鹼指示劑3.2.4 Applications of acid-base titration酸鹼滴定的應用3.3 Complexometric titration配位滴定3.3.1 Metal-chelate complexes金屬螯合物3.3.2 EDTA 乙二胺四乙酸3.3.3 EDTA titration curves EDTA滴定曲線3.3.4 Metal Ion indicators金屬離子指示劑3.3.5 Applications of EDTA titration techniques EDTA滴定方法的應用3.4 Oxidation-reduction titration氧化還原滴定3.4.1 Redox reactions氧化還原反應3.4.2 Rate of redox reactions氧化還原反應的速率3.4.3 Titration curves滴定曲線3.4.4 Redox indicators氧化還原指示劑3.4.5 Applications of redox titrations氧化還原滴定的應用3.5 Precipitation titration沉澱滴定3.5.1 Precipitation reactions沉澱滴定反應3.5.2 Titration curves滴定曲線3.5.3 End-point detection終點檢測ProblemsTerms to understand重點內容概述Chapter 4 Potentiometry 電位分析法4.1 Introduction簡介4.1.1 Classes and characteristics分類及性質4.1.2 Definition定義4.2 Types of potentiometric electrodes電極種類4.2.1 Reference electrodes 參比電極4.2.2 Indicator electrodes指示電極4.2.3 Electrode response and selectivity電極響應及選擇性4.3 Potentiometric methods and application電位法及應用4.3.1 Direct potentiometric measurement 直接電位法4.3.2 Potentiometric titrations電位滴定4.3.3 Applications of potentiometry 電位法應用ProblemsTerlns to understand重點內容概述Chapter 5 Chromatography色譜法5.1 An introduction to chromatographic methods色譜法概述5.2 Fundamental theory of gas chromatography氣相色譜基本原理5.2.1 Plate theory塔板理論5.2.2 Kinetic theory(rate theory) 速率理論5.2.3 The resolution Rs as a measure of peak separation 分離度5.3 Gas chromatography 氣相色譜5.3.1 Components of a gas chromatograph 氣相色譜儀的組成5.3.2 Stationary phases for gas-liquid chromatography 氣液色譜固定相5.3.3 Applications of gas-liquid chromatography 氣液色譜的應用5.3.4 Adsorption chromatography 吸附色譜5.4 High performance liquid chromatography 高效液相色譜5.4.1 Instrumentation 儀器組成5.4.2 High-performance partition chromatography 高效分配色譜5.5 Miscellaneous separation methods 其他分離方法5.5.1 High-performance ion-exchange chromatography 高效離子交換色譜5.5.2 Capillary electrophoresis 毛細管電泳5.5.3 Planar chromatography 平板色譜ProblemsTerms to understand重點內容概述Chapter 6 Atomic Absorption Spectrometry原子吸收光譜分析法6.1 Introduction 概述6.2 Principles 原理6.2.1 The process of AAS，resonance line and absorption line 原子吸收光譜法的過程，共振線及吸收線6.2.2 The number of ground atom and the temperature of flame 基態原子數與光焰溫度6.2.3 Quantitative analysis of AAS原子吸收光譜定量分析6.3 Instrumentation 儀器6.3.1 Primary radiation sources 光源6.3.2 Atomizer 原子儀器6.3.3 Optical dispersive systems 分光系統6.3.4 Detectors 檢測器6.3.5 Signal measurements 信號測量6.4 Quantitative measurements and interferences 定量測定及干擾6.4.1 Quantitative measurements 定量測定6.4.2 Interferences 干擾6.4.3 Sensitivity6.5 Applications of AAS原子吸收光譜法的應用ProblemsTerms to understand重點內容概述Chapter 7 Ultraviolet and Visible Spectrophotometry 紫外-可見分光光度法7.1 Introduction簡介7.2 Ultraviolet and visible absorption spectroscopy 紫外-可見吸收光譜7.2.1 Introduction for radiant energy 輻射能簡介7.2.2 Selective absorption of radiation and absorbance spectrum 物質對光的選擇性吸收和吸收光譜7.2.3 Absorbing species and electron transition 吸收物質與電子躍遷7.3 Law of absorption吸收定律7.3.1 Lambert-Beer's law朗伯-比爾定律7.3.2 Absorptivity吸光係數7.3.3 Apparent deviations from Beer's law對比爾定律的明顯偏離7.4 Instruments儀器7.5 General types of spectrophotometer分光光度計種類7.6 Application of UV-Vis absorption spectroscopy 紫外-可見吸收光譜的應用7.6.1 Application of absorption measurement to qualitative analysis 光吸收測定在定性分析上的應用7.6.2 Quantitative analysis by absorption measurements 光吸收測量定量分析法7.6.3 Derivative spectrophotometry 導數分光光度法ProblemsTerms to understand重點內容概述Chapter 8 Infrared Absorption Spectroscopy紅外吸收光譜8.1 Theory of infrared absorption紅外吸收基本原理8.1.1 Dipole changes during vibrations and rotations 振轉運動中的偶極距變化8.1.2 Mechanical model of stretching vibrations 伸縮振動機械模型8.1.3 Quantum treatment of vibrations 振動的量子力學處理、8.1.4 Types of molecular vibrations分子振動形式8.2 Infrared instrument components紅外儀器組成8.2.1 Wavelength selection波長選擇8.2.2 Sampling techniques 採樣技術8.2.3 Infrared spectrophotometers for qualitative analysis 定性分析用紅外分光光度計8.2.4 Other techniques其他技術8.3 The group frequencies of functional groups in organic compounds 有機化合物官能團的特徵頻率8.4 The factors affecting group frequencies 影響基團特徵吸收頻率的因素8.4.1 Adjacent groups 鄰近基團的影響。

欧氏空间中常高阶平均曲率紧致凸超曲面与高斯映像

欧氏空间中常高阶平均曲率紧致凸超曲面与高斯映像王琪【摘要】For an oriented, compact and convex hypersurface M without boundary in the ( n+1 )-dimensional Euclidean space Rn+1, we apply a known integral formula and put out a new skill to prove that if there exits an integer r (1≤r≤n-1) such that the r-mean curvature Hr is a constant and if the Gauss map of M is a topological homeomorphism onto the standard unit sphere Sn, then M is totally umbilical.%针对(n+1)维欧氏空间Rn+1中紧致无边凸超曲面M,利用一个已知的积分公式,并提出一种新的技巧,证明了:如果存在一个整数r(1≤r≤n-1)使得M的第r阶高阶平均曲率Hr是常数,并且M的高斯映照是到标准单位球面Sn的拓扑同胚,则M全脐.【期刊名称】《福州大学学报（自然科学版）》【年(卷),期】2018(046)003【总页数】4页(P307-310)【关键词】欧氏空间;凸超曲面;高阶平均曲率;高斯映照;全脐性质【作者】王琪【作者单位】贵阳学院数学与信息科学学院,贵州贵阳 550005【正文语种】中文【中图分类】O186.160 引言2004年， Alencar等[1]研究球面空间Sn+1中常高阶平均曲率的紧致无边超曲面，建立起一类积分公式，并通过这些公式，利用高斯映像来刻画超曲面的全脐性质得到了定理1. 文献[2]研究欧氏空间Rn+1中以低维球面为边界, 具有常数高阶平均曲率的紧致超曲面. 其建立了相应的积分公式，同时利用高斯映照的一种条件，讨论超曲面的性质，得到了定理2. 本研究讨论欧氏空间Rn+1紧致无边超曲面, 通过一个已知的积分公式，并采用一种新的“分割”技巧，利用高斯映照来刻画超曲面的性质，得到了一个新的定理，即以下定理3.1 预备知识设n维黎曼流形M等距浸入到一个(n+1)维黎曼流形中，并设λi(1≤i≤n)是M的主曲率函数，则M第r阶高阶平均曲率Hr有以下定义[1-4]:同时定义H0≡1.当M可定向时， M具有整体的单位法向量场N. 记Sn是n维标准单位球面，则M的Gauss映照定义为φ:M→Sn，φ(p)=N(p) (∀p∈M)需要使用一个积分公式，即下列引理1.引理1 [4] 设M是Rn+1中紧致无边定向超曲面. 用x表示M在Rn+1中的位置向量， N表示M的光滑单位法向量场. 则以下积分公式成立其中：〈, 〉表示Rn+1的欧氏内积， dx表示黎曼流形M的n维黎曼体积元.2 主要结果定理1 [1] 设M是单位球面Sn+1空间中紧致无边超曲面. 假设M的某一个高阶平均曲率Hr(1≤r≤n-1)是正常数，同时下列不等式处处成立Hr-1≥0， H1Hr-1≥Hr>0如果M的高斯映照像落在一个闭的半球面内，则M是全脐的.定理2[2] 设M是Rn+1中定向紧致带边界超曲面，且假设M的某个高阶平均曲率Hr(2≤r≤n)是常数，同时M⊂π的边界为圆球面Sn-1. 如果M的高斯映照像落在一个超平面π的一侧，则M一定是n维圆盘或球面盖.定理3 设M是Rn+1中紧致无边定向凸超曲, 且M的某一个高阶平均曲率Hr(1≤r≤n-1)是常数. 如果M的高斯映照是到标准单位球面Sn的拓扑同胚，则M是全脐的.证明因为M是凸的，所以M整个落在其每一点之切超平面的一侧，则可以适当选择M的单位法向量场的指向，那么在M的每一点处，其主曲率函数λi(1≤i≤n)均为非负.因为M是紧致无边的，根据文献[4]的论证， M必有严格椭圆点，即：在该点处必有Hr>0.而本研究假设Hr是常数，所以有Hr>0 (∀x∈M)根据文献[1-2]，产生代数不等式结合以上两式有Hi>0，∀x∈M (i=1, 2, …, r)根据另外一个熟知的代数不等式[1-8]，可得∀x∈M (i=1, 2, …, r)(1)由以上分析立即有H1≥H2/H1≥…≥Hr/Hr-1≥Hr+1/Hr (∀x∈M)(2)现在，应用引理1的积分公式，有(3)(4)因为Hr为常数，由式子(3)可得(5)再由式子(4)～(5)可知(6)注意到式子(2)则有H1Hr-Hr+1≥0 (∀x∈M)(7)为完成证明，本研究将采用“分割法”：即分部分来证明M的脐性.首先，因为M是凸的和闭的，所以M必定包围一个(n+1)维的凸区域D. 则M 恰是某个凸区域D的边界，即M=∂D.任意取定D的一个内点O∈D，并且以点O作为Rn+1的原点.任意取定Rn+1的一个通过原点的n维超平面n，即Σn是Rn+1的一个线性子空间. 同时记Sn是n维标准单位球面.记Σn-1=Sn∩Σn=Sn-1则Σn-1=Sn-1即是Sn的一个(n-1)维赤道，而Σn就是相应的n维赤道面.因为Σn经过Rn+1的原点，所以可以假定Σn={x=(x1, …, xn, xn+1)∈Rn+1:xn+1=0}今写M的单位法向量场N为N=(τ1, …, τn, τn+1)并记于是坐标平面Σn将单位球面Sn划分为上、中、下三个部分，即：而且互不相交.注意到本研究的假设：高斯映射φ:M→Sn是拓扑同胚，若记则M的这三个部分互不相交，而且M=M+∪M-∪M0现在，可以将式子(6)重新写作(8)其中以下分别估计上面三项： A、 B和C.首先，因为假设高斯映射φ:M→Sn是拓扑同胚，而拓扑映射保持维数不变，所以M0=φ-1(Sn-1)是Rn+1中一个(n-1)维子集. 于是有(9)其次，因为高斯映射φ:M→Sn是拓扑同胚，所以，将落在n维坐标面Σn的同一侧，否则将破坏φ-1的连续性. 再注意到：事实上有M0=φ-1(Sn-1)=M∩Σn，于是的位置向量x与其单位法向量N(x)的夹角必须是锐角，所以〈x, N〉=〈x, N(x)〉>0 (∀x∈M+)(10)由式子(7)和式子(10)，有(11)另一方面，只能落在n-维坐标面Σn的另外一侧，因此有〈x, N〉=〈x, N(x)〉>0 (∀x∈M-)(12)由式子(7)和式子(12)，可得(13)现在，由式子(6)、 (9)以及式子(12)～(13)，事实上有A=B=0(14)再次注意到式子(7)、 (9)以及式子(11)、 (13)，可得H1Hr-Hr+1=0, ∀x∈M+ (∀x∈M-)(15)至此，由式子(15)可知：在M+和M-的每一点处，式子(7)的不等式，事实上都取到了等号.再由式子(1)中任意一个不等号取到等号的条件，可得： M+和M-的每一点都是脐点.最后，因为Rn+1的坐标原点可以在M所包围的凸区域的内部任意取定，而且Sn的赤道平面Σn也是可以任意旋转的，所以M是全脐的.参考文献：[1] ALENCAR H, ROSENBERG H, STANTOS W. On the Gauss map of hypersurfaces with constant scalar in spheres[J]. Proceedings of the American Mathematical Society, 2004, 132(12): 3731-3739.[2] 张远征. Rn+1中常高阶平均曲率超曲面[J]. 数学学报， 2005, 48(4):647-652.[3] HARDY G, LITLEWOOD J, POLYA G. Inequalities[M]. Cambridge: Cambridge University Press, 1989.[4] KHO S E. A characterization of round spheres[J]. Proceedings of the American Mathematical Society, 1998, 126(12): 3657-3660.[5] 王琪. 正曲率空间形式中超曲面的全脐性与高阶平均曲率[J]. 数学学报, 2014, 57(1): 47-50.[6] 王琪. anti de Sitter 空间中全脐类空超曲面与高阶平均曲率[J]. 山西大学学报(自然科学版), 2016, 39(3): 403-405.[7] 王琪. de Sitter空间中紧致类空超曲面的全脐性与高阶平均曲率[J]. 安徽大学学报(自然科学版), 2016, 40(1): 7-10.[8] 王琪. 双曲空间中全脐超曲面与高斯映照像[J]. 浙江大学学报(理学版), 2016, 43(5): 537-538， 549.。

Double Bubbles Minimize

a r X i v :m a t h /0003157v 2 [m a t h .D G ] 10 O c t 2000Annals of Mathematics,151(2000),459–515Double bubbles minimize By Joel Hass and Roger Schlafly *Abstract The classical isoperimetric inequality in R 3states that the surface of small-est area enclosing a given volume is a sphere.We show that the least area surface enclosing two equal volumes is a double bubble,a surface made of two pieces of round spheres separated by a ﬂat disk,meeting along a single circle at an angle of 120◦. 1.Introduction In this paper we ﬁnd the unique surface of smallest area enclosing two equal volumes.The surface is called a double bubble ,and is made of two pieces of round spheres separated by a disk,meeting along a single circle at an angle of 120◦.This is the form assumed by two equally sized soap bubbles which are brought together until their boundaries conglomerate to form a common wall.See Figure 1,due to J.Sullivan.Isoperimetric problems,which study maximizing the size of an enclosed bounded region whose boundary is of ﬁxed size,are among the oldest problems in mathematics.For a broad discussion of isoperimetric problems see Osserman [33].The two volume isoperimetric problem in R 3was considered by the Belgian physicist J.Plateau [35],and appears in C.V.Boys’famous book on soap bubbles.As Boys wrote,“When however the bubble is not single,say two have been blownin real contact with one another,again the bubbles must togethertake such a form that the total surface of the two spherical segmentsand of the part common to both,which I shall call the interface,isthe smallest possible surface which will contain the two volumes ofair and keep them separate.”460JOEL HASS AND ROGER SCHLAFLYFigure1.A double bubble enclosing two equal volumes.We now outline brieﬂy the recent history of the problem,with more details given later. F.Almgren proved the existence of bubble clusters enclosing a given collection of volumes in three-space and minimizing area among a general class of surfaces.J.Taylor established a regularity result for these bubble clusters,showing that they satisfy regularity properties physically observed by Plateau. B.White suggested an argument for showing that a two-region minimizer must be a surface of revolution,and J.Foisy and M.Hutchings ﬁlled in the details of this argument.At this point it was known that the minimizer consisted of pieces of Delaunay surfaces meeting at120◦.Hutchings went on to eliminate many remaining conﬁgurations,leaving just the standard double bubble and an additional family known as torus bubbles as the potential minimizers.In this paper we closely examine the family of torus bubbles and show they cannot be minimizers for the equal volume double bubble problem. Each torus bubble is determined by an ODE,and we can rigorously eliminate the entire collection by an examination of the solutions of these ODEs using a computational technique called interval arithmetic,described in Section5.At the end,only the standard double bubble remains as a possible minimizer.Extensive interest in the problem was generated in recent years by work of F.Morgan.The planar case has been solved in[1]by methods special to two dimensions.A fundamental paper of Almgren[3]established the existence of solutions to a great variety of geometric minimization problems,including multiple component isoperimetric problems in R3.Despite the fact that exis-tence was established some time ago,our result is theﬁrst explicit example of a surface in R3solving a multiple regionﬁnite volume isoperimetric problem. Lawlor and Morgan have shown that the cone over the regular tetrahedron is a minimizing surface among all those with its boundary that separate the four faces of the tetrahedron,but it is still not known whether this surface is minimizing without the separation condition.DOUBLE BUBBLES MINIMIZE461 Such multiple region isoperimetric problems arise in manyﬁelds,including the growth and shape of biological cells[28],[40].There were extensive stud-ies of such problems by physical and biological scientists in the19th century [24],[35].Plateau established experimentally that a soap bubble cluster is a piecewise-smooth surface having only two types of singularities.Theﬁrst type of singularity occurs when three smooth surfaces come together along a smooth triple curve at an angle of120◦.The second type of singularity occurs when six smooth surfaces and four triple curves converge at a point.The angles are then equal to those of the cone over the1-skeleton of a regular tetrahedron.A mathematical proof that these types of singularities are the only ones possible in a minimizing bubble in R3was given by Taylor[39].However there were no explicit minimizing bubbles known for any collection of volumes exhibiting either of these singularities.Thus we have found theﬁrst explicit example of a closed minimizing surface in R3known to exhibit some of the singularities predicted by Plateau.The arguments presented in this paper are a mixture of geometrical anal-ysis and of estimates of geometric quantities obtained by the use of numerical computation.We perform these calculations with strict estimates on the accu-racy of the computations.Since it is still somewhat unusual in a mathematical proof to use digital computers to do calculations involving real numbers,we will say a few words about the nature of this part of the argument.Computers are widely known in the mathematical community to have been used successfully for analysis of discrete and combinatorial problems, such as the4-color theorem[6],[37].Problems such as isoperimetric inequali-ties are qualitatively diﬀerent,since they inherently involve real numbers.Real numbers are represented in digital computers in aﬂoating point format which allows exact description of only aﬁnite number of rationals.Simple calcula-tions,like division,or even addition,lead to unrepresentable numbers,and so the computer must round oﬀ.For purposes of mathematical proof,the size of the round-oﬀmust be accurately tracked throughout the calculation.Methods for strictly estimating solutions of diﬀerential equations exist, but are not yet widely used in the mathematical community.The vast ma-jority of numerical work is for approximation and simulation that does not meet the standards of a mathematical proof.However there have been some important results achieved through rigorous use ofﬂoating point numerical methods to achieve traditional mathematical proofs,notably Lanford’s work on the Feigenbaum Conjectures[27]and Feﬀerman and de la Llave’s work on the stability of matter[12],[15]and the work of McKay and Percival on nonexistence of invariant tori[34].See also recent work of Hales on the Ke-pler Conjecture[19]and Gabai,Meyerhoﬀand Thurston[17]on hyperbolic puter calculations are essential to our proof that equal vol-462JOEL HASS AND ROGER SCHLAFLYume double bubbles minimize area.There are too many calculations to be done by hand.The proof parameterizes the space of possible solutions by a two-dimensional rectangle,one dimension corresponding to an angle and the other to a mean curvature.This rectangle is subdivided into15,016smaller rectangles which are investigated by calculations involving a total of51,256numerical integrals. Every calculation is done with strict error bounds,and all results are precise mathematical statements.All operations conform to the IEEE754standard for computer arithmetic,a widely adopted standardized method for implementing real number(ﬂoating point)computations on computers[5].Our methods in-dicate that numerical techniques are likely to play an important role in future geometrical arguments.The main result is the following,announced in[20]:Theorem1.The unique surface of least area enclosing two equal vol-umes in R3is a double bubble.Corollary2.For any surface in R3enclosing two regions,each of volume v,the area a satisﬁesa3≥243πv2,with equality if and only if the surface is a double bubble enclosing two regions of volume v.Theﬁrst issue in establishing Theorem1is toﬁnd an appropriate category of surface in which to minimize area.It suﬃces in this paper to consider piecewise smooth two-dimensional surfaces.It is sometimes useful to consider a much more general notion of surface,such as the(F,ε,δ)sets described in[3].Our arguments actually imply that the double bubble minimizes in this larger class;see Theorem2.2. Consideration of such a larger class of surfaces is needed primarily in the establishment of the existence and regularity of a minimizer,carried out in [3]and[39],and we will not need to be overly concerned with it in this paper.Deﬁne a piecewise-smooth curve to be an embeddedﬁnite union of smooth curves,with any two either disjoint or having intersection contained in their endpoints.Deﬁne a piecewise-smooth surface to be an embeddedﬁnite union of smooth surfaces with piecewise-smooth boundary curves,with any two sur-faces either disjoint or intersecting along piecewise-smooth curves contained in their boundaries.The set of points which are not in the interior of a smooth subsurface of a piecewise-smooth surface is called the singular set.Deﬁne a bubble to be a piecewise-smooth surface satisfying:1.Each two dimensional surface has constant mean curvature.DOUBLE BUBBLES MINIMIZE4632.The singular set is of the type described by Plateau.It consists of smoothtriple curves along which three smooth surfaces come together at an angleof120◦and isolated vertices where six smooth surfaces and four triplecurves converge at a point.The angles at the point are equal to those ofthe cone over the1-skeleton of a regular tetrahedron.3.The mean curvatures around an edge in the singular set where threesurfaces have common boundary sum up to zero.The above conditions are necessary for no local perturbation of a piecewise-smooth surface to decrease the area while preserving the volume in each of itscomplementary regions.A bubble enclosing regions of prescribed volumes is called a minimizingbubble if it minimizes area among all bubbles enclosing the same volumes.The regions are not necessarily connected.Given positive constants v1and v2,let a(v1,v2)denote the inﬁmum of thearea of piecewise-smooth surfaces enclosing two regions R1and R2in R3whichare closed bounded sets with disjoint interiors such that volume(R1)=v1andvolume(R2)=v2.We will refer to the interior of the complement of R1∪R2 as the exterior region R0.Note that R0may not be connected,in which casethe bubble encloses some compact“empty regions”.A double bubble enclosing volumes v1,v2is a surface made of three pieces ofround spheres,meeting along a single circle at an angle of120◦,and enclosingtwo connected regions having volumes v1and v2.We consider the plane tobe a sphere of inﬁnite radius in this setting,allowing the interface of a doublebubble to be aﬂat disk,as occurs when v1and v2are equal.See Figure2.In this and all otherﬁgures,we assume that the axis of rotational symmetryis the x-axis.Furthermore,we will always take the generating curves for asurface of revolution to be in the upper half of the xy-plane.It has been conjectured since the work of Plateau that double bubbles givethe most eﬃcient shape for enclosing two given volumes.Conjecture3.The double bubble enclosing volumes v1,v2is the uniquesurface having area equal to a(v1,v2).Theorem1solves this conjecture in the case that v1=v2.Deep results of Almgren and Taylor,summarized in Section2,imply thatfor any two positive numbers v1,v2there exists a minimizing bubble S(v1,v2)in R3which encloses volumes v1and v2.Arguments based on ideas of B.White and F.Morgan,and developed in[16],[22],[32],show that any solution must be a surface of revolution.We will need to refer to this proof,so we present a simple version for the case of two regions in R3in Theorem2.6.The lack of such an argument for isoperimetric problems involving three or more regions in R3makes those problems more formidable.464JOEL HASS AND ROGER SCHLAFLYWhile the reduction to a surface of revolution gives an enormous simpli-ﬁcation in the scale of the problem,F.Morgan has pointed out several major topological obstacles to solving the double bubble conjecture.Theﬁrst is that the regions R1and R2bounded by the minimizing bubble S(v1,v2)may not be connected,as illustrated in Figure3.A second problem is that S(v1,v2)may enclose bounded regions in R3 which do not form part of either R1or R2,as in Figure4.These regions are called empty regions by Morgan.Figure2.Equal and unequal volume double bubbles in cross-section.The axis of revolution is the x-axis.Figure3.Cross-section of a bubble with a nonconnected region.The nonconnected region is shaded.Figure4.Cross-section of a bubble with an empty torus region.The shaded region generates a solid torus when revolved around the x-axis.DOUBLE BUBBLES MINIMIZE465A third problem is that S(v1,v2)may enclose nonsimply connected re-gions.Since the surface is a surface of revolution,these regions are homeomor-phic to solid tori,and we call them torus components.There are therefore numerous possible conﬁgurations which a minimizing bubble might take.A recent breakthrough due to Hutchings[22]has given restrictions on the type of surfaces that can arise in the double bubble problem, eliminating many possibilities.Some remaining possibilities are depicted in Figure5.For the case of equal volumes,Hutchings showed that there were further constraints.Each region must be connected,leaving only two possible conﬁgurations,the double bubble and an additional class of possibilities,called torus bubbles,whose properties will be discussed in Section4.Figure5.Cross-sections of some possible bubble conﬁgurations.In Section5we describe the algorithm used in a series of computations which show that torus bubbles are not minimizers for the two equal volume isoperimetric problem in R3.Techniques extending those developed in this paper also prove Conjec-ture3for other volume ratios,but at this time do not seem to suﬃce to cover all cases.2.Existence and regularityAlmgren showed in[3]that there exists an area minimizing surface S(v1,v2) in R3among the set of surfaces enclosing volumes v1,v2.Here surface refers to a generalized notion deﬁned using the methods of geometric measure theory, speciﬁcally what Almgren calls(F,ε,δ)sets.For our purposes it suﬃces that this class includes the piecewise-smooth surfaces.466JOEL HASS AND ROGER SCHLAFLYTheorem2.1([3]).An area minimizing surface S(v1,v2)exists and is a smooth surface almost everywhere.The nature of the singularities of S(v1,v2)was established by Taylor.Theorem2.2([39]).S(v1,v2)is a piecewise-smooth surface.Its singu-larities consist of smooth triple curves along which three smooth surfaces come together at an angle of120◦and isolated points where four triple curves and six pieces of surface converge.At these isolated points the asymptotic cone is the cone over the1-skeleton of a regular tetrahedron.Thus Taylor’s work established that S(v1,v2),which Almgren’s theorem established as a minimizer among a generalized class of surfaces called(F,ε,δ) surfaces,is a bubble,in our terminology.It follows that if we can establish the minimality of the double bubble among our class of bubbles then we will also show it minimizes among the more general class of surfaces considered by Alm-gren.We now establish some properties of minimizing bubbles.Lemma2.3, Corollary2.4and Lemma2.5,are standard results in variational geometry. We include brief proofs for completeness.Lemma2.3states that the sum of the(oriented)mean curvatures of all points crossed by a path starting and ending in the same region is zero.It implies that the sum of the(oriented) mean curvatures of all points crossed by any path depend only on its starting and ending points.These results hold even if the path crosses through many surfaces,and if it starts and ends in diﬀerent components of the same region.Lemma2.3.Letγbe an oriented curve in R3intersecting a minimizing bubble B transversely at regular points such that the initial andﬁnal points of γlie in the interior of the same region.Then the sum of the mean curvatures of all the points ofγ∩B,oriented byγ,is zero.Proof.Perturb the curve slightly so that each of its intersections with B becomes perpendicular.Consider a deformation of B which pushes points in R3nearγa uniform distance along the curve.Toﬁrst order this preserves the volume of each region.The derivative of the area,toﬁrst order,is given by the sum of the mean curvatures over the points ofγ∩B.If this sum is nonzero,a deformation can be deﬁned which decreases area while preserving the volume of each region.A special case of the above lemma occurs whenγis a simple closed curve encircling a triple curve of the bubble.The lemma then implies that the sum of the mean curvatures around the triple curve adds up to zero.This local minimization condition is built into our deﬁnition of a bubble.More generally, the lemma implies that any two surfaces separating the same pair of regions have the same mean curvature.DOUBLE BUBBLES MINIMIZE467 Corollary2.4.If S1and S2are surfaces in a bubble,each of which separate regions R1and R2,then their mean curvatures are equal.Proof.Consider an arcαwhich starts in R1passes through S1into R2, then transversely through the bubble and on through S2back into R1.A subarcβofαstarts and ends in R2.Applying Lemma2.3to bothαandβimplies the corollary.Lemma2.5.A minimizing bubble B is connected.Proof.If not,we can move a component by an isometry of R3until it touches a distinct component.The resulting singularity violates those allowed in Theorem2.2.The following result wasﬁrst observed by B.White and F.Morgan,and ﬁrst written down by Foisy in[16].Since we will need to refer to the proof, we present an argument for the case of two regions in R3.More detailed and general arguments can be found in the work of Hutchings[22]and Morgan[32].Theorem2.6.S(v1,v2)is a surface of revolution.Proof.Since S(v1,v2)is a minimizing bubble,each of its faces separates a pair of distinct regions-either separating R1from R2or one of them from R0(which is not connected if there are empty regions.)Given any unit vector Z in R3,there is a plane P Z in R3perpendicular to Z which bisects the total volume v1+v2enclosed by S(v1,v2).If Q Z is a parallel plane which also bisects the total volume,then P Z and Q Z split the regions R1 and R2in the same proportions.(In fact P Z must coincide with Q Z,but we do not need to show this.It is also irrelevant to this argument whether S(v1,v2) contains empty regions.)Let f(Z)denote the proportion of the volume of R1on the side of P Z to which Z points.The function f(Z)is well deﬁned, continuous on the unit2-sphere,0≤f(Z)≤1and f(−Z)=1−f(Z).Along any great circle of directions on the unit2-sphere,the Intermediate Value Theorem implies that there are at least two points where f(Z)=1/2and the plane P Z bisects the volume of both regions.Weﬁx Z to be such a vector.Consider the intersection of S(v1,v2)with each of the two half-spaces determined by P Z.The intersection of a face of S(v1,v2)with the plane P Z is given by a graph in P Z with smooth edges and isolated vertices,or else the face is completely contained in P Z,since the face is a smooth constant mean curvature surface[18].Reﬂection of the smaller area piece of S(v1,v2)lying in one of these half-spaces gives a new surface S1(v1,v2)which encloses regions of the same volumes.Since S1(v1,v2)cannot have less area,it must have the same area as S(v1,v2).S1(v1,v2)has the property that reﬂection through P Z468JOEL HASS AND ROGER SCHLAFLYpreserves the new regions,which we continue to call R0,R1and R2.If the plane P Z contained a face of S(v1,v2)then this face would now separate two components of the same region in S1(v1,v2),and area could be reduced by removing that face from S1(v1,v2),preserving the volume of each region.So P Z cannot contain a face of S(v1,v2).We repeat the above argument along a great circle of the unit2-sphere consisting of the unit vectors which are perpendicular to Z,toﬁnd a plane P W perpendicular to P Z which bisects the volume of the regions R1and R2 in S1(v1,v2).This process constructs a surface S2(v1,v2),still having the same area as S(v1,v2),enclosing regions of the same volumes,and for which reﬂection through each of P Z and P W preserves R1and R2.Since composing reﬂections through two perpendicular planes gives a ro-tation of angle180◦,it follows that rotation of angle180◦about L=P Z∩P W preserves the regions R1and R2.Now consider any plane Q containing L. Note that half of the volume of R1and R2lies on each side of Q.As before the plane Q cannot contain a face of S2(v1,v2).If the intersection of Q with S2(v1,v2)is not perpendicular to Q,then replacing half of S2(v1,v2)with its reﬂected image through Q gives a new minimizing surface which is not smooth, and has singularities not allowed by Theorem2.2.Thus S2(v1,v2)is perpen-dicular to each plane through L and is therefore a surface of revolution around L.S1(v1,v2)and S2(v1,v2)may not a priori be identical,but they coincide on a half-space of R3.Moreover this half-space was chosen arbitrarily,so that S1(v1,v2)is cut by P W into two pieces,each half of some surface of revolution. The axis of each of these surfaces of revolution is L=P Z∩P W,so they co-incide and S1(v1,v2)itself is a surface of revolution.Similar reasoning shows that S(v1,v2)is a surface of revolution,and coincides with S2(v1,v2)every-where.If we make S1by reﬂecting the other half of S across P Z,then the same argument shows that the other half of S is also a surface of revolution about a line L′in P Z.The Almgren and Taylor regularity results,and in particular the unique continuation property of constant mean curvature surfaces,imply that L′=L,so S itself is a surface of revolution about L.We next summarize some key results obtained by Hutchings in[22].Recall that S(v1,v2)is a minimizing bubble that separates R3into regions R1of volume v1and R2of volume v2and that a(v1,v2)is the area of S(v1,v2).Theorem2.7.The function a(v1,v2)is strictly concave on[0,+∞)×[0,+∞).Corollary2.8.If v′1>v1then a(v′1,v2)>a(v1,v2).Proof.As v1→+∞,a(v1,v2)≥a(v1)→+∞.Concavity of the function a implies the corollary.Corollary2.9.S(v1,v2)has no empty regions.Proof.If there is an empty region,join it it to one of the other regions by removing a face on its boundary and apply the previous corollary.Deeper connectedness results also follow from concavity.In particular, Hutchings deduced that each of R1and R2is connected if the volumes are equal.Theorem2.10.S(v1,v1)encloses exactly two connected components.Proof.See Hutchings[22,Th.4.2].3.Delaunay surfacesIn this section,we summarize the classiﬁcation theory of constant mean curvature surfaces of revolution,the Delaunay surfaces,and present some prop-erties of these surfaces that will be used in our study of bubbles.The mean curvature of a surface in R3is the trace of the second fundamen-tal form of the puting this requires choosing a unit normal vector ﬁeld to the surface.The mean curvature vectorﬁeld,formed by scaling a unit normal vectorﬁeld by the mean curvature,gives the direction of a variation which decreases the area of a surface as quickly as possible[38].For a surface of revolution we have simple formulas for the mean curvature. Consider a surface of revolution about the x-axis in R3,with generating curve contained in the upper half-plane.If the generating curve is a graph,y=y(x), then the mean curvature h,equal to the sum of the principal curvatures,is given by the formulas:k m=−¨yyabove if the curve is oriented left-to-right,or equivalently if the graph’s tan-gent vector has a positive x-component.If its tangent vector has a negative x-component,the signs in the formulas for k m and k p should be reversed in the above formulas.For curves with vertical tangent vectors the above formulas do not apply directly,but in our applications vertical tangents are isolated points, and the limits of the above formulas will give k m and k p at vertical tangencies.Surfaces of revolution having constant mean curvature wereﬁrst studied by Euler,and classiﬁed by Delaunay[13].The term Delaunay surface is used to refer to these surfaces.Theorem3.1.The surfaces of revolution of constant mean curvature are the plane,sphere,catenoid,cylinder,unduloid,and nodoid.The generating curves of a nodoid and an unduloid,called the nodary and the undulary,are periodic along the x-axis,and have exactly one local minimum and one local maximum in each period.The undulary is a graph over the x-axis.The nodary has one local maximum,one local minimum and two vertical tangencies in each period.Proof.Expositions of the classiﬁcation of Delaunay surfaces can be found in[14]and[23].The nodoid and unduloid are not as well known as the other surfaces.An undulary can be obtained by rolling an ellipse along the x-axis and tracing the path taken by one of the foci.A nodary can be obtained by rolling a hyperbola [14].See Figure6.Figure6.Portions of a nodary and an undulary,generating curves of a nodoid and an unduloid.With their generating curves oriented left to right,the mean curvature of each of the Delaunay surfaces is nonnegative,with the exception of the nodoid.The sign of the mean curvature of a nodoid depends upon the choice of orientation for its generating nodary,and can be positive or negative.If a surface of revolution minimizes area among all surfaces of revolution surrounding a given volume,then its generating curve satisﬁes an associated Euler-Lagrange equation.This second order ODE implies that the mean cur-vature is constant[14].The mean curvature of the surface of revolution generated by a C2graph y(x)is given by:(1)h=k m+k p=−¨y y(1+˙y2)3/2−11+˙y2=2hy˙y+2y˙y¨ydxwhere f(x)is the function(2)f(x)=hy2−2y1+˙y2.Therefore f is constant along the graph.The fact that the equation for con-stant mean curvature surfaces of revolution has aﬁrst integral was known to Plateau[35,pp.138–139],who in turn references work of Beer.Korevaar, Kusner and Solomon called f the force of a constant mean curvature surface, and pointed out that it has a physical interpretation as the net force exerted by a soapﬁlm on a plane cutting oﬀan end of the surface.They also showed that it could be deﬁned for more general constant mean curvature surfaces[25],where it serves as a useful analytical tool.A convenient way of expressing the force of a constant mean curvature surface of revolution which holds even when the generating curve is not a graph is given by the formula(3)f=hy2−2y cosα.Hereαdenotes the angle between the positive x-axis and the generating curve of the surface of revolution,which is an oriented curve.This makes sense for all values ofα.In our applications,we alway orient the generating curve of a Delaunay surface,and thusﬁx an unambiguous sign for the mean curvature.Lemma3.2.Given h,x0,y0>0and˙y0,there exists a unique Delaunay curveδthrough the point(x0,y0)having slope˙y0and generating a Delaunay surface D having mean curvature h.The slope˙y0can be inﬁnite.The Delaunay surface is a sphere S if and only if(4)h=h S=21+˙y20.If h>h S,then D is a nodoid andδlies strictly below the circleσgen-erating S in a neighborhood of(x0,y0).An arc ofδwhich is decreasing as it leaves(x0,y0)remains beneathσas long as it remains a graph over the x-axis.If h<h S,δlies strictly aboveσin a neighborhood of(x0,y0).If0<h< h S then D is an unduloid or cylinder,if h<0then D is a nodoid,and if h=0 then D is a catenoid.Proof.Solving equation1for¨y gives a second order ODE satisﬁed by the generating curves of Delaunay surfaces at points where˙y isﬁnite:1+˙y2(5)¨y=−h(1+˙y2)3/2+。

微分几何中的英文单词和短语

Chapter 3 Intrinsic Geometry of Surfaces
共变导数
covariant derivative
高斯方程
Gauss equation
曲面局部理论的 fundamental theorem of local theory
基本定理
of surface
高斯-博内定理 Gauss-Bonnet theorem
超曲面空间旋轮线高斯曲率伸展曲面方向导数黑塞矩阵椭圆点
hyper surface space cycloid Gauss curvature elongated surface directional derivative Hessian matrix elliptic point
第三章曲面的内蕴几何
第四章常曲率空间
Chapter 4 Spaces of Constant Curvature
双曲空间截空间弧长的变差最短路径旋转群四元数空间双二面体群等度量变换等距表示
hyperbolic space truncated space variation of arc length shortest path rotation group quaternion space double dihedral group isometric transformation isometric representation
正方环面黎曼流形切线空间
square torus Riemannian manifold tangent space
第五章黎曼流形
Chapter 5 Riemannian Manifold
凯莱映射链规则复流形可数性公理曲率张量共变张量对偶基爱因斯坦空间爱因斯坦张量克莱因瓶拉普拉斯算子李导数测度张量距离张量射影平面旋转矩阵

南开大学光学工程专业英语重点词汇汇总

光学专业英语部分refraction [rɪˈfrækʃn]n.衍射reflection [rɪˈflekʃn]n.反射monolayer['mɒnəleɪə]n.单层adj.单层的ellipsoid[ɪ'lɪpsɒɪd]n.椭圆体anisotropic[,ænaɪsə(ʊ)'trɒpɪk]adj.非均质的opaque[ə(ʊ)'peɪk]adj.不透明的；不传热的；迟钝的asymmetric[,æsɪ'metrɪk]adj.不对称的；非对称的intrinsic[ɪn'trɪnsɪk]adj.本质的，固有的homogeneous[,hɒmə(ʊ)'dʒiːnɪəs;-'dʒen-] adj.均匀的；齐次的；同种的；同类的，同质的incidentlight入射光permittivity[,pɜːmɪ'tɪvɪtɪ]n.电容率symmetric[sɪ'metrɪk]adj.对称的；匀称的emergentlight出射光；应急灯.ultrafast[,ʌltrə'fɑ:st,-'fæst]adj.超快的；超速的uniaxial[,juːnɪ'æksɪəl]adj.单轴的paraxial[pə'ræksɪəl]adj.旁轴的；近轴的periodicity[,pɪərɪə'dɪsɪtɪ]n.[数]周期性；频率；定期性soliton['sɔlitɔn]n.孤子，光孤子；孤立子；孤波discrete[dɪ'skriːt]adj.离散的，不连续的convolution[,kɒnvə'luːʃ(ə)n]n.卷积；回旋；盘旋；卷绕spontaneously:[spɒn'teɪnɪəslɪ] adv.自发地；自然地；不由自主地instantaneously:[,instən'teinjəsli]adv.即刻；突如其来地dielectricconstant[ˌdaiiˈlektrikˈkɔnstənt]介电常数，电容率chromatic[krə'mætɪk]adj.彩色的；色品的；易染色的aperture['æpətʃə;-tj(ʊ)ə]n.孔，穴；（照相机，望远镜等的）光圈，孔径；缝隙birefringence[,baɪrɪ'frɪndʒəns]n.[光]双折射radiant['reɪdɪənt]adj.辐射的；容光焕发的；光芒四射的; photomultiplier[,fəʊtəʊ'mʌltɪplaɪə]n.[电子]光电倍增管prism['prɪz(ə)m]n.棱镜；[晶体][数]棱柱theorem['θɪərəm]n.[数]定理；原理convex['kɒnveks]n.凸面体；凸状concave['kɒnkeɪv]n.凹面spin[spɪn]n.旋转；crystal['krɪst(ə)l]n.结晶，晶体；biconical[bai'kɔnik,bai'kɔnikəl] adj.双锥形的illumination[ɪ,ljuːmɪ'neɪʃən] n.照明；[光]照度；approximate[ə'prɒksɪmət] adj.[数]近似的；大概的clockwise['klɒkwaɪz]adj.顺时针方向的exponent[ɪk'spəʊnənt；ek-] n.[数]指数；even['iːv(ə)n]adj.[数]偶数的；平坦的；相等的eigenmoden.固有模式；eigenvalue['aɪgən,væljuː]n.[数]特征值cavity['kævɪtɪ]n.腔；洞，凹处groove[gruːv]n.[建]凹槽，槽；最佳状态；惯例；reciprocal[rɪ'sɪprək(ə)l]adj.互惠的；相互的；倒数的，彼此相反的essential[ɪ'senʃ(ə)l]adj.基本的；必要的；本质的；精华的isotropic[,aɪsə'trɑpɪk]adj,各向同性的；等方性的phonon['fəʊnɒn]n.[声]声子cone[kəʊn]n.圆锥体，圆锥形counter['kaʊntə]n.柜台；对立面；计数器；cutoff['kʌt,ɔːf]n.切掉；中断；捷径adj.截止的；中断的cladding['klædɪŋ]n.包层；interference[ɪntə'fɪər(ə)ns]n.干扰，冲突；干涉borderline['bɔːdəlaɪn]n.边界线，边界；界线quartz[kwɔːts]n.石英droplet['drɒplɪt]n.小滴，微滴precision[prɪ'sɪʒ(ə)n]n.精度，[数]精密度；精确inherently[ɪnˈhɪərəntlɪ]adv.内在地；固有地；holographic[,hɒlə'ɡræfɪk]adj.全息的；magnitude['mægnɪtjuːd]n.大小；量级；reciprocal[rɪ'sɪprək(ə)l]adj.互惠的；相互的；倒数的，彼此相反的stimulated['stimjə,letid]v.刺激（stimulate的过去式和过去分词）cylindrical[sɪ'lɪndrɪkəl]adj.圆柱形的；圆柱体的coordinates[kəu'ɔ:dineits]n.[数]坐标；external[ɪk'stɜːn(ə)l;ek-]n.外部；外观；scalar['skeɪlə]n.[数]标量；discretization[dɪs'kriːtaɪ'zeɪʃən]n.[数]离散化synthesize['sɪnθəsaɪz]vt.合成；综合isotropy[aɪ'sɑtrəpi]n.[物]各向同性；[物]无向性；[矿业]均质性pixel['pɪks(ə)l;-sel]n.（显示器或电视机图象的）像素（passive['pæsɪv]adj.被动的spiral['spaɪr(ə)l]n.螺旋；旋涡；equivalent[ɪ'kwɪv(ə)l(ə)nt]adj.等价的，相等的；同意义的; transverse[trænz'vɜːs;trɑːnz-;-ns-]adj.横向的；横断的；贯轴的;dielectric[,daɪɪ'lektrɪk]adj.非传导性的；诱电性的;n.电介质；绝缘体integral[ˈɪntɪɡrəl]adj.积分的；完整的criteria[kraɪ'tɪərɪə]n.标准，条件（criterion的复数）Dispersion:分散|光的色散spectroscopy[spek'trɒskəpɪ]n.[光]光谱学photovoltaic[,fəʊtəʊvɒl'teɪɪk]adj.[电子]光电伏打的，光电的polar['pəʊlə]adj.极地的；两极的；正好相反的transmittance[trænz'mɪt(ə)ns;trɑːnz-;-ns-] n.[光]透射比；透明度dichroic[daɪ'krəʊɪk]adj.二色性的；两向色性的confocal[kɒn'fəʊk(ə)l]adj.[数]共焦的；同焦点的rotation[rə(ʊ)'teɪʃ(ə)n]n.旋转；循环，轮流photoacoustic[,fəutəuə'ku:stik]adj.光声的exponential[,ekspə'nenʃ(ə)l]adj.指数的;fermion['fɜːmɪɒn]n.费密子（费密系统的粒子）semiconductor[,semɪkən'dʌktə]n.[电子][物]半导体calibration[kælɪ'breɪʃ(ə)n]n.校准；刻度；标度photodetector['fəʊtəʊdɪ,tektə]n.[电子]光电探测器interferometer[,ɪntəfə'rɒmɪtə]n.[光]干涉仪；干涉计static['stætɪk]adj.静态的；静电的；静力的;inverse相反的，反向的，逆的amplified['æmplifai]adj.放大的；扩充的horizontal[hɒrɪ'zɒnt(ə)l]n.水平线，水平面；水平位置longitudinal[,lɒn(d)ʒɪ'tjuːdɪn(ə)l;,lɒŋgɪ-] adj.长度的，纵向的；propagate['prɒpəgeɪt]vt.传播；传送；wavefront['weivfrʌnt]n.波前；波阵面scattering['skætərɪŋ]n.散射；分散telecommunication[,telɪkəmjuːnɪ'keɪʃ(ə)n] n.电讯；[通信]远程通信quantum['kwɒntəm]n.量子论mid-infrared中红外eigenvector['aɪgən,vektə]n.[数]特征向量；本征矢量numerical[njuː'merɪk(ə)l]adj.数值的；数字的ultraviolet[ʌltrə'vaɪələt]adj.紫外的；紫外线的harmonic[hɑː'mɒnɪk]n.[物]谐波。

RevModPhys.82.331

CONTENTS
I. The Cosmological Standard Model A. Historical outline B. Friedmann models 1. The metric 2. Redshift and expansion 3. The radiation-dominated phase 4. Age, distances, and horizons C. Structures 1. Structure growth 2. The power spectrum 3. Nonlinear evolution II. The Age of the Universe A. Nuclear cosmochronology 1. The age of the Earth 2. The age of the Galaxy B. Stellar ages C. Cooling of white dwarfs D. Summary III. The Hubble Constant A. Hubble constant from Hubble’s law 1. Hubble’s law 2. The distance ladder 3. The HST key project B. Gravitational lensing C. Summary IV. Big-Bang Nucleosynthesis A. The origin and abundance of 4He 1. Elementary considerations 2. The Gamow criterion 3. Elements produced 4. Helium abundance 5. Expected abundances and abundance trends B. Observed element abundances 1. Principles 2. Evolutionary corrections 3. Speciﬁc results 332 332 333 333 333 334 335 335 335 336 337 337 337 337 338 339 340 340 340 340 340 341 342 342 343 343 343 343 344 344 345 345 346 346 346 346

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∂U C
Hale Waihona Puke g(η, V ) = 0(1)
where ν is the unit normal vector ﬁeld of ∂ U in Λ and η is the unit normal vector ﬁeld of C in M . This formula can now applied to the approximate solution of the CMC perturbation problem, having mean curvature approximately equal to h, in the following way. Choose the open set U as one of the building blocks of the approximate solution. Then ∂ U consists of a disjoint union of small (n − 1)-spheres at the centres of the necks attaching U to its neighbours, and C is the disjoint union of the small disks that cap these spheres oﬀ. The left hand side of (1) now encodes information about the width and location of the neck regions of U . If the left hand side of (1) is suﬃciently close to zero, then one says that U is in balanced position. The idea is now that in order to be able to overcome the obstructions to the solvability of the constant mean curvature equations, the approximate solution must be constructed in such a way that all its building blocks are in balanced position. The balancing condition amounts to a form of local symmetry satisﬁed by each building block with respect to its nearest neighbours in the initial conﬁguration that is to be glued together. This is similar to what happens in Euclidean space. However, force balancing in itself is not 2
arXiv:0707.2069v1 [math.DG] 13 Jul 2007
Abstract The techniques developed by Butscher in [4] for constructing constant mean curvature (CMC) hypersurfaces in Sn+1 by gluing together spherical building blocks are generalized to handle less symmetric initial conﬁgurations. The outcome is that the approximately CMC hypersurface obtained by gluing the initial conﬁguration together can be perturbed into an exactly CMC hypersurface only when certain global geometric conditions are met. These balancing conditions are analogous to those that must be satisﬁed in the ‘classical’ context of gluing constructions of CMC hypersurfaces in Euclidean space, although they are more restrictive in the Sn+1 case. An example of an initial conﬁguration is given which demonstrates this fact; and another example of an initial conﬁguration is given which possesses no symmetries at all.
spheres. Although it is easy to imagine how to use the gluing technique in ambient manifolds other than R3 , provided one has enough simple building blocks, it is not clear that the gluing technique will be quite a successful, in particular when the ambient manifold is compact. In Butscher’s and Butscher-Pacard’s work [2, 3, 4], the gluing technique for constructing CMC hypersurfaces has been successfully adapted to work in the compact ambient manifold Sn+1 . In these papers, the CMC building blocks of the sphere — namely the hyperspheres obtained by intersecting Sn+1 with hyperplanes and the product spheres of the form Sp (cos(α)) × Sq (sin(α)) for α ∈ (0, π/2) called the generalized Cliﬀord tori — are conﬁgured in a variety of diﬀerent ways, glued together using small embedded catenoidal necks, and perturbed into CMC hypersurfaces. One should imagine that the hypersurfaces constructed in these papers are analogues of the ‘classical’ constructions that are possible in Euclidean space. As before, there are obstructions for solving the CMC equation on an arbitrary initial conﬁguration; and when certain global geometric conditions are met, the obstructions disappear. These geometric conditions are indeed close analogues of the balancing conditions identiﬁed by Kapouleas; but the conditions seem to be stronger in the Sn+1 case than in the Euclidean case. This is to be expected since Sn+1 is compact and the additional requirement that the initial conﬁgurations must close should have ramiﬁcations in the analysis of the CMC equation. The balancing condition is best explained in the more general context found in Korevaar-KusnerSolomon’s work [9]. First, suppose that Λ is a hypersurface with constant mean curvature h in an (n + 1)-dimensional Riemannian manifold (M, g) possessing a Killing ﬁeld V . Let U be an open ¯ be an open set in M such that ∂ U ¯ = ∂ U ∪ C where C is a bounded n-dimensional set in Λ and U cap which may have multiple components. Then the ﬁrst variation formula for the n-volume of U ¯ in the direction of the variation subject to the constraint of constant enclosed (n + 1)-volume of U determined by V implies g(ν, V ) − h