A theorem of H. Hopf and the Cauchy-Riemann inequality II

This is a sequel to “A theorem of H. Hopf and the Cauchy-Riemann inequality” [AdCT]. Here the result of the previous paper is extended (see the precise statement in Section 1 of the present paper) to surfaces in three-dimensional homogeneous Riemannian manifolds whose group of isometries has dimension four and the bundle curvature is nonzero, whereas in the previous paper only the case of vanishing bundle curvature was treated. * Partially supported by MEC-FEDER Grant No. MTM 2004-00160.     

单位球面中具平行单位平均曲率向量的子流形

设M是n-维闭黎曼流形,等距浸入(n+p)-维单位球空间S^n+p,具有平行的单位平均曲率向量。若S≤min{2n/3,2(n-1)^1/2},其中S是M的第二基本形式长度的平方,则M是S^n+p的一个(n+1)-维全测地子流形Sⁿ⁺¹中的超曲面。     

On a kähler version of cheeger-gromoll-perelman's soul theorem

In this note, we will prove a Kähler version of Cheeger-Gromoll-Perelman's soul theorem, only assuming the sectional curvature is nonnegative and bisectional curvature is positive at one point.     

Asymptotic behavior of spreading fronts in the anisotropic Allen–Cahn equation on Rn

We consider the Cauchy problem for the anisotropic (unbalanced) Allen–Cahn equation on ${\mathbb{R}}^n$ with $n\ge 2$ and study the large time behavior of the solutions with spreading fronts. We show, under very mild assumptions on the initial data, that the solution develops a well-formed front whose position is closely approximated by the expanding Wulff shape for all large times. Such behavior can naturally be expected on a formal level and there are also some rigorous studies in the literature on related problems, but we will establish approximation results that are more refined than what has been known before. More precisely, the Hausdorff distance between the level set of the solution and the expanding Wulff shape remains uniformly bounded for all large times. Furthermore, each level set becomes a smooth hypersurface in finite time no matter how irregular the initial configuration may be, and the motion of this hypersurface is approximately subject to the anisotropic mean curvature flow $V_{\gamma }={\kappa }_{\gamma }+c$ with a small error margin. We also prove the eventual rigidity of the solution profile at the front, meaning that it converges locally to the traveling wave profile everywhere near the front as time goes to infinity. In proving this last result as well as the smoothness of the level surfaces, an anisotropic extension of the Liouville type theorem of Berestycki and Hamel (2007) for entire solutions of the Allen–Cahn equation plays a key role.     

On a theorem by Hsiang and Yu

In 1841, Delaunay classified surfaces of revolution with constant mean curvature in the Euclidean three space. As a byproduct of his result, one obtains: A surface of revolution has a periodic generating curve if and only if its mean curvature is non-zero. One hundred and forty years after Delaunay’s work, Hsiang and Yu extended this result to higher dimensions, by extending Delaunay’s idea of tracing the locus of a focus by rolling a given conic section along a line. In this paper, we give a new proof of their result using elementary ODE theory to obtain the periodicity of the solutions under consideration.      

圆球面为边的常F-平均曲率类空超曲面

设F(t)是上伪球面H_+~n上时轴旋转不变且满足凸条件的光滑函数.本文证得:Lorentz空间L~(n+1)中,以圆球面S~(n-1)(R)为边界且有非零常F-平均曲率的紧致类空超曲面必为Wulff帽.     

Material voids in elastic solids with anisotropic surface energies

This work discusses the role of highly anisotropic interfacial energy for problems involving a material void in a linearly elastic solid. Using the calculus of variations it is shown that important qualitative features of the equilibrium shape of the void may be deduced from smoothness and convexity properties of the interfacial energy.     

A sphere theorem for odd-dimensional submanifolds of spheres

We establish a topological sphere theorem from the point of view of submanifold geometry for odd-dimensional submanifolds of a unit sphere. We give examples which show that our result is optimal. Moreover, we note the assumption that the dimension is odd is essential.

     

An algebraic version of Tamano's theorem for countably compact spaces

We show that if X is countably compact but not compact then one can find a compact space K such that X⊕K does not embed closedly into any normal topological group.     

Mean width inequalities for symmetric Wulff shapes

We establish the mean width inequalities for symmetric Wulff shapes by a direct approach.We also yield the dual inequality along with the equality conditions.These new inequalities have Barthe’s mean width inequalities for even isotropic measures and its dual form as special cases.     

正曲率子流形的几何刚性定理

§1Introduction LetMnbeann-dimensionalcompactRiemannianmanifoldisometricallyimmersedinto an(n+p)-dimentionalcompleteandsimplyconnectedRiemannianmanifoldFn+p(c)with constantcurvaturec.DenotebyKMandHthesectionalcurvatureandmeancurvatureofM respectively.In[10],Yauprovedthefollowingstrikingresult.TheoremA.LetMnbeann-dimensionalorientedcompactminimalsubmanifoldin Sn+p(1).IfthesectionalcurvatureofMisnotlessthanp-12p-1,thenMiseitherthetotally geodesicsphere,thestandardimmersionoftheproductoftw…     

A global pinching theorem for surfaces with constant mean curvature in

Let be a compact immersed surface in the unit sphere with constant mean curvature . Denote by the linear map from into , , where is the linear map associated to the second fundamental form and is the identity map. Let denote the square of the length of . We prove that if , then is either totally umbilical or an -torus, where is a constant depending only on the mean curvature .

     

An optimal rigidity theorem for complete submanifolds in a sphere

It is proved that if M^n is an n-dimensional complete submanifold with parallel mean curvature vector and flat normal bundle in S^n＋p（1）, and if supM S 〈 α（n, H）, where α（n,H）=n＋n^3/2（n-1）H^2-n（n-2）/n（n-1）√n^2H^4＋4（n-1）H^2,then M^n must be the totally urnbilical sphere S^n（1/√1＋H^2）.An example to show that the pinching constant α（n, H） appears optimal is given.     

局部对称共形平坦黎曼流形中紧致子流形的一个刚性定理

本文研究局部对称共形平坦黎曼流形N^n p(p≥2）中具有平行平均曲率向量的紧致子流形M^n的余维可约性问题，在n≥8的条件下得到了最佳拼挤常数。     

Anisotropic eigenvalues upper bounds for hypersurfaces in weighted Euclidean spaces

We prove anisotropic Reilly-type upper bounds for divergence-type operators on hypersurfaces of the Euclidean space in presence of a weighted measure.     

Convex billiards and a theorem by E. Hopf

Supported by the Colton Fellowship     

First stability eigenvalue characterization of CMC Hopf tori into Riemannian Killing submersions

We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature orientable surfaces immersed in a Riemannian Killing submersion. As a consequence, the strong stability of such surfaces is studied. We also characterize constant mean curvature Hopf tori as the only ones attaining the bound in certain cases.     

A Bernstein theorem for complete spacelike hypersurfaces of constant mean curvature in Minkowski space

In this paper we prove a general Bernstein theorem on the complete spacelike constant mean curvature hypersurfaces in Minkowski space. The result generalizes the previous result of Cao-Shen-Zhu (1998) and Xin (1991). The proof again uses the fact that the Gauss map of a constant mean curvature hypersurface is harmonic, which was proved by K. T. Milnor (1983), and the maximum principle of S. T. Yau (1975).

     

Embedded Hypersurfaces in Sn+1 with Constant Mean Curvature and Spherical Boundary

It is proved that an embedded hypersurface in a hemisphere of the Euclidean unit spherewith constant mean curvature and spherical boundary inherits, under certainconditions, the symmetries of its boundary. In particular, spherical caps are theonly such hypersurfaces whose boundary are geodesic spheres.