On a generalized 1-harmonic equation and the inverse mean curvature flow

We introduce and study generalized 1-harmonic equations (1.1). Using some ideas and techniques in studying 1-harmonic functions from Wei (2007)  [1], and in studying nonhomogeneous 1-harmonic functions on a cocompact set from Wei (2008)  [2, (9.1)], we find an analytic   quantity w$w$ in the generalized 1-harmonic equations  (1.1) on a domain in a Riemannian n$n$-manifold that affects the behavior of weak solutions of  (1.1), and establish its link with the geometry   of the domain. We obtain, as applications, some gradient bounds and nonexistence results for the inverse mean curvature flow, Liouville theorems for p$p$-subharmonic functions of constant p$p$-tension field, p≥n$p\ge n$, and nonexistence results for solutions of the initial value problem of inverse mean curvature flow.     

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

In this paper, the complete spacelike hypersurface with constant normal scalar curvature in a locally symmetric Lorentz space is discussed. Several classified theorems are obtained using the operator _L1$L_1$ introduced by S.Y. Cheng and S.T. Yau (1977) [4].     

Bernstein theorems for space-like graphs with parallel mean curvature and controlled growth

In this paper, we obtain an Ecker–Huisken-type result for entire space-like graphs with parallel mean curvature.     

Isotropic submanifolds of pseudo-Riemannian spaces

The family of all the submanifolds of a given Riemannian or pseudo-Riemannian manifold is large enough to classify them into some interesting subfamilies such as minimal (maximal), totally geodesic, Einstein, etc. Most of these have been extensively studied by many authors, but as far as we know, no paper has hitherto been published on the class of isotropic submanifolds. The purpose of this paper is therefore to gain a better understanding of this interesting class of submanifolds that arise naturally in mathematics and physics by studying their relationships with other closely distinguished families.     

On Randers spaces of constant flag curvature

This paper concerns a ubiquitous class of Finsler metrics on smooth manifolds of dimension n. These are the Randers metrics. They figure prominently in both the theory and the applications of Finsler geometry. For n ≥ 3, we consider only those with constant flag curvature. For n = 2, we focus on those whose flag curvature is a (possibly constant) function of position only. We characterize such metrics by three efficient conditions. With the help of examples in 2 and 3 dimensions, we deduce that the Yasuda-Shimada classification of Randers space forms actually addresses only a special case. The corrected classification for that special case is sharp, holds for n ≥ 2, and follows readily from our three necessary and sufficient conditions.     

Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle

We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in a toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states) and some non-monotone Lagrangian torus fibers.     

On surfaces whose twistor lifts are harmonic sections

We study surfaces whose twistor lifts are harmonic sections, and characterize these surfaces in terms of their second fundamental forms. As a corollary, under certain assumptions for the curvature tensor, we prove that the twistor lift is a harmonic section if and only if the mean curvature vector field is a holomorphic section of the normal bundle. For surfaces in four-dimensional Euclidean space, a lower bound for the vertical energy of the twistor lifts is given. Moreover, under a certain assumption involving the mean curvature vector field, we characterize a surface in four-dimensional Euclidean space in such a way that the twistor lift is a harmonic section, and its vertical energy density is constant.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Isolation Phenomenon for Complete Surfaces Arising from Yang–Mills Theory

Let M be a complete surface with parallel mean curvature in a complete simply connected space form F ^2+p (c) of constant curvature c. Denote by H and S the mean curvature and the squared length of the second fundamental form of M respectively. Motivated by L ²-isolation phenomenon in Yang–Mills theory, we prove that if , where c + H ² > 0, D(H,c) is an explicit positive constant depending on H and c, then , i.e., M is a totally umbilical sphere . Research supported by the Chinese NSF, Grant No. 10231010; Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.     

Minimal hypersurfaces in a nearly Kaehler 6-sphere

In this paper we show that for a compact minimal hypersurface M$M$ of constant scalar curvature in the unit sphere S⁶$S^6$ with the shape operator A$A$ satisfying ‖A‖²>5${\Vert A\Vert }^2>5$, there exists an eigenvalue λ>10$\lambda >10$ of the Laplace operator of the hypersurface M$M$ such that ‖A‖²=λ−5${\Vert A\Vert }^2=\lambda -5$. This gives the next discrete value of ‖A‖²${\Vert A\Vert }^2$ greater than 0 and 5.     

On the mean curvature of spacelike submanifolds in semi-Riemannian manifolds

In this paper we establish some estimates for the higher-order mean curvature of a complete spacelike hypersurface in spacetimes with sectional curvature satisfying certain condition. We also obtain the estimate for the mean curvature of a complete spacelike submanifold in semi-Riemannian space forms.     

On symplectic submanifolds of cotangent bundles

Necessary and sufficient conditions are given for a symplectic submanifold of a cotangent bundle to itself be a cotangent bundle.Partially supported by NSF grant DMS-9222241.     

Normalization and prescribed extrinsic scalar curvature on lightlike hypersurfaces

In a recent paper [C. Atindogbé, Scalar curvature on lightlike hypersurfaces, Appl. Sci. 11 (2009) 9–18], the present author considered the concept of extrinsic (induced) scalar curvature on lightlike hypersurfaces. This scalar quantity has been studied on lightlike hypersurfaces equipped with a given normalization. But a very important problem was left open: How to characterize the set of all normalizations admitting a prescribed extrinsic scalar curvature? In this paper, we provide various responses to this question, supported by examples.     

Biminimal properly immersed submanifolds in the Euclidean spaces

We consider a complete nonnegative biminimal   submanifold M$M$ (that is, a complete biminimal submanifold with λ≥0$\lambda \ge 0$) in a Euclidean space E^N${\mathbb{E}}^N$. Assume that the immersion is proper  , that is, the preimage of every compact set in E^N${\mathbb{E}}^N$ is also compact in M$M$. Then, we prove that M$M$ is minimal. From this result, we give an affirmative partial answer to Chen’s conjecture. For the case of λ<0$\lambda <0$, we construct examples of biminimal submanifolds and curves.