常数平均曲率子流形的曲率估计及其应用

本文利用活动标架法及Ｌａｐｌａｃｉａｎ特征值方法研究了空间形中具有常数平均曲率的子流形，给出了高斯曲率与数量曲率的一种估计方法，证明了空间形中具有常数平均曲率的子流形上一个单连通有界区域为稳定的两个充分条件．     

Finsler空间上的Weyl曲率

The Weyl curvature of a Finsler metric is investigated. This curvature constructed from Riemannain curvature. It is an important projective invariant of Finsler metrics. The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature. A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved. It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.     

Some Inequalities for the Lp-p olar Curvature Images of Star Bo dies

Zhu, L¨u and Leng extended the concept of Lp-polar curvature image. We con-tinuously study the Lp-polar curvature image and mainly expound the relations between the volumes of star bodies and their Lp-polar curvature images in this article. We first establish the Lp-a?ne isoperimetric inequality associated with Lp-polar curvature image. Secondly, we give a monotonic property for Lp-polar curvature image. Finally, we obtain an interesting equation related to Lp-projection body of Lp-polar curvature image and Lp-centroid body.     

On surfaces of prescribed weighted mean curvature

Utilising a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature.     

Volume-preserving flow by powers of the mth mean curvature

We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the mth mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere.     

Total Extrinsic Curvature of Certain Distributions on Closed Spaces of Constant Curvature

An integral formula for symmetric functions of curvature ofdistributions on closed constant nonnegative sectional curvature spacesis proved. The distributions under consideration are orthogonal to atotally geodesic foliation and the main theorem extends a previousresult concerning the total curvature of codimension-one foliations.     

Characterizations of Null Holomorphic Sectional Curvature of G C R-Lightlike Submanifolds of Indefinite Nearly K ¨ahler Manifolds

We obtain the expressions for sectional curvature, holomorphic sectional curvature and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite nearly K ¨ahler manifold and obtain characterization theorems for holo-morphic sectional and holomorphic bisectional curvature. We also establish a condi-tion for a GCR-lightlike submanifold of an indefinite complex space form to be a null holomorphically flat.     

Solitons of curvature

An intristic geometry of surfaces is discussed. In geodesic coordinates the Gauss equation is reduced to the Schrödinger equation where the Gaussian curvature plays the role of a potential. The use of this fact provides an infinite set of explicit expressions for the curvature and metric of a surface. A special case is governed by the KdV equation for the Gaussian curvature. We consider the integrable dynamics of curvature via the KdV equation, higher KdV equations and (2+1)-dimensional integrable equations with breaking solitons.     

On the curvatures of Einstein spaces

For a pseudo-Riemannian manifold (M, g) of dimensionn3, we introduce a scalar curvature functionS(V) for non-degenerate subspacesV ofT _pM which is a generalization of the scalar curvature, and give some characterizations of Einstein spaces in terms of this scalar curvature function. We also give a characterization for spaces of constant curvature. As an application of our results, we show that the Ricci curvature or the sectional curvature of a Lorentz manifold is constant if the scalar curvature function for non-degenerate subspaces is bounded.Partially supported by the grants from TGRC.     

Estimates and nonexistence of solutions of the scalar curvature equation on noncompact manifolds

This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.     

Distributional radius of curvature

We show that any continuous plane path that turns to the left has a well‐defined distribution that corresponds to the radius of curvature of smooth paths. We show that the distributional radius of curvature determines the path uniquely except for a translation. We show that Dirac delta contributions in the radius of curvature correspond to facets, that is, flat sections of the path, and show how a path can be deformed into a facet by letting the radius of curvature approach a delta function. Copyright © 2005 John Wiley & Sons, Ltd.     

On a natural algebraic affine connection on the space of almost complex structures and the curvature of teichmüller space with respect to its Weil-Petersson metric

A formula for the sectional curvature of Teichmüller space with respect to the Weil-Petersson metric is derived in terms of the Laplace-Beltrami operator on functions. It will be shown that the sectional curvature as well as the holomorphic sectional curvature and Ricci curvature are negative. Bounds on the holomorphic and the Ricci curvature are given.     

On the Ricci Curvature of Compact Spacelike Hypersurfaces in de Sitter Space

In this paper we establish a sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a pinching condition for the Ricci curvature. Our result will be a consequence of an integral formula involving the Ricci curvature and the scalar curvature of the hypersurface. We also derive some other consequences and applications of this formula.     

Compact Manifolds with Positive Einstein Curvature

In this paper we study positive Einstein curvature which is a condition on the Riemann curvature tensor intermediate between positive scalar curvature and positive sectional curvature. We prove some constructions and obstructions for positive Einstein curvature on compact manifolds generalizing similar well known results for the scalar curvature. Finally, because our problem is relatively new, many open questions are included.     

Existence of constant mean curvature 2-Spheres in Riemannian 3-spheres

We prove the existence of branched immersed constant mean curvature (CMC) 2-spheres in an arbitrary Riemannian 3-sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3-sphere is positively curved. To achieve this, we develop a min-max scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a bi-harmonic approximation procedure to obtain compactness of the new functional, a derivative estimate of the min-max values to gain energy upper bounds for min-max sequences for almost every choice of mean curvature, and a Morse index estimate to obtain another uniform energy bound required to reach the remaining constant mean curvatures in the presence of positive curvature.     

Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.     

关于凸体的Lp-曲率映象的不等式

Lutwak提出了凸体的Lp-曲率映象的概念,并证明了凸体与其Lp-曲率映象的体积之间的一个不等式．本文给出了Lutwak结果的一个一般形式,继而证明了凸体与其Lp-曲率映象的极的体积之间的一个不等式,并得到了凸体的Lp-投影体和Lp-曲率映象的体积之间的一个不等式．     

Generalized Solution of a Kind of Nonparametric Curvature Evolution with Boundary Condition

The existence and uniqueness of the generalized solution for a kind of nonparametric curvature flow problem are obtained. This kind of curvature flow problem describes the evolution of graphs with speed depending on the reciprocal of the Gauss curvature.     

On the nonexistence of a hypersurface of prescribed mean curvature with a given boundary

A general method is developed for finding necessary conditions for a given codimension-two submanifold Γ of a riemannian manifold to be the boundary of an immersed hypersurface of prescribed mean curvature. In the simplest case the condition is a comparison of the magnitude of the mean curvature with the ratio of the volume of the projection of Γ into a hyperplane to the volume of the interior of that projection. The method is applied to show that certain recent existence results for surfaces of prescribed mean curvature may not be quantitatively improved.