Convex Hypersurfaces with Prescribed Curvature and Boundary in Riemannian Manifolds

In this article a priori estimates at the boundary for the second fundamental form of n-dimensional convex hypersurfaces M with prescribed curvature quotient Sn (M)/Sl (M) in Riemannian manifolds are derived. A consequence of these estimates and other known results is an existence theorem for such hypersurfaces, which is a generalization of a recent result of Ivochkina and Tomi to the Riemannian case.     

Starshaped Compact Hypersurfaces with Prescribed M-th Mean Curvature in Elliptic Space

We consider the problem of finding a compact starshaped hypresurface in a space form for which the normalized m-th elementary symmetric function of principal curvatures is a prescribed function. In this paper the conditions for the existence of at least one solution to a nonlinear second order elliptic equation of that problem are established in case of a space form with positive sectional curvature.     

Prescribed Mean Curvature Hypersurfaces in Hn+1(-1) with Convex Planar Boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hⁿ⁺¹(-1). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.     

Inverse Anisotropic Curvature Flow from Convex Hypersurfaces

In this paper, we study the inverse anisotropic curvature flow from strictly convex hypersurfaces. We show the long-time existence and the convergence to the Wulff shape after rescaling, under certain conditions on the general speed functions.     

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.     

Contraction Flow by Certain Curvature of Convex Hypersurfaces

In the following mixed tensors on the hypersurface M Rn+1 under consideration willbe denoted by T={ Tijkl} ,the induced metric by g={ gij} and the second fundamentalform by A={ hij} .We always sum overrepeated indices from1 to n and use brackets forthe inner product on M:〈Tijk,Sijk〉 =gisgjαgkβTijk Ssαβ,　 | T| 2 =〈Tijk,Tijk〉.Setrkl =gkαhαl,Sm(λ1 ,… ,λn) = i1<… 相似文献   

Curvature Flow of Complete Convex Hypersurfaces in Hyperbolic Space

We investigate the existence, convergence, and uniqueness of modified general curvature flow (MGCF) of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.     

局部对称黎曼流形中具有常中曲率完备超曲面

该文研究了局部对称黎曼流形中的具有常平均曲率完备超曲面,获得了超曲面的一个特征定理,此定理推广了一些已有的结论.     

局部对称Lorentz空间中的具有常数量曲率的完备类空超曲面

The complete space-like hypersurfaces with constant normal saclar curvature is discussed in a locally symmetric Lorentz space. A classified theorem is obtained by the operator L_1 introduced by S Y Cheng and S T Yau [3].     

局部对称流形中具常平均曲率的完备超曲面

本文研究了局部对称流形中具常平均曲率的完备超曲面，得到了这种超曲面的一个特征定理，推广了H．Alencar和M．do Carmo以及N．X．Shui和G．Q．Wu的结果。     

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M ^m of a Euclidean space R ^m+p is said to have harmonic mean curvature vector field if ${\Delta \vec{H}=0}$ , where ${\vec{H}}$ is the mean curvature vector field of ${M\hookrightarrow R^{m+p}}$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.     

由曲率函数和外力场之差支配的凸超曲面的发展

考虑由曲率函数和外力场之差支配的凸超曲面的发展.证明了外力场为常向量场时,初始超曲面的凸性是保持的,且曲率流在有限时间内爆破.对于线性外力场,初始超曲面的凸性保持.而且,若线性常数为负数,则曲率流在有限时间内收敛到一点;若线性常数为正数且初始曲率小于某一与外力场有关的常数,则曲率流光滑地存在于任意有限时间区间,并发散到无穷;若线性常数为正数且初始曲率大于某一与外力场有关的常数,则曲率流在有限时间内爆破.     

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.     

Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature

The main result of this paper states that the traceless second fundamental tensor A⁰ of an n-dimensional complete hypersurface M, with constant mean curvature H and finite total curvature, _M |A⁰|ⁿ dv_M < , in a simply-connected space form (c), with non-positive curvature c, goes to zero uniformly at infinity. Several corollaries of this result are considered: any such hypersurface has finite index and, in dimension 2, if H ² + c > 0, any such surface must be compact.     

A Note on Doubly Connected Surfaces of Constant Mean Curvature with Prescribed Boundary

In this paper we prove the existence of a constant mean curvature surface spanning two given convex curves in parallel planes of ℝ³ under hypotheses relating the distance between the planes, the curvature of the curves and the mean curvature. It is also proved that the surface is a radial graph over a unit sphere.This author was partially Supported by Fapergs. Mathematics Subject Classifications (2000): 53A10, 53C42, 35J60     

Radial Graphs of Constant Mean Curvature and Doubly Connected Minimal Surfaces with Prescribed Boundary

In this paper we investigate existence and uniqueness of radial graphs ofconstant mean curvature (cmc) with prescribed boundary. Our main resultestablishes the existence of a minimal radial anullus spanning two givenconvex curves in parallel planes of R³; we also obtain a variant ofa well-known result of James Serrin about the existence of radial cmc graphsover convex domains in the sphere.     

Rotational Hypersurfaces with Constant Gauss-Kronecker Curvature

The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in Rⁿ. They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several geometric properties of such hypersurfaces. In particular, they discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume, which can be seen as the higher-dimensional generalization of the pseudo-sphere.     

On the Prescribed Boundary Mean Curvature Problem via Local Pohozaev Identities

This paper deals with the following prescribed boundary mean curvature problem in B^N ■ where ■ is a bounded nonnegative function with ■ .Combining the finite-dimensional reduction method and local Pohozaev type of identities,we prove that N≥5 and ■ has a stable critical point ■ with r₀> d and ■>0,then the above problem has infinitely many solutions,whose energy can be made arbitrarily large.Here our result fill the gap that the above critical points may inlude the sad...