Finsler manifolds with nonpositive flag curvature and constant S-curvature

The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in Rⁿ with negative flag curvature and constant S-curvature. In this paper, we are going to show a global rigidity theorem that every Finsler metric with negative flag curvature and constant S-curvature must be Riemannian if the manifold is compact. We also study the nonpositive flag curvature case.supported by the National Natural Science Foundation of China (10371138).     

Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL ₁-small integral anisotropy haveL _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in theW _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.     

单位球中完备超曲面的一个刚性定理

本文研究了单位球中的数量曲率满足r=aH+b的完备超曲面的问题.利用极值原理的方法,获得了超曲面的一个刚性结果,推广了这一类具有常中曲率或者常数量曲率超曲面的结果.     

Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

We investigate the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and prove that if M ⁿ is simply connected and the k-th Ricci curvature of M ⁿ is bounded below by a quantity involving the mean curvature of M ⁿ and the curvature of the ambient manifold, then M ⁿ is diffeomorphic to the standard sphere ${\mathbb{S}^n}$ . For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold M ⁿ is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.     

The existence of k-convex hypersurface with prescribed mean curvature

Using the strong maximum principle, we obtain a constant rank theorem for the k-convex solutions of semilinear elliptic partial differential equations. As an application we obtain an existence theorem of k-convex starshaped hypersurface with prescribed mean curvature in R ⁿ⁺¹.     

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L ² harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.     

On the Topology of Vacuum Spacetimes

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n + 1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold Sⁿ \Sigma^n to an asymptotically Euclidean solution of the constraints on \mathbbRⁿ \mathbb{R}^n . For any Sⁿ \Sigma^n which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [16], which is restricted to constant mean curvature data.     

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e ^2u=K_o(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore ^2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem. Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755. Research supported in part by NSF Grant DMS-9877077     

Complete noncompact manifolds with harmonic curvature

Let (M ⁿ , g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (M ⁿ , g) is a space form if it has sufficiently small L ^n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M ⁿ , g) with positive scalar curvature.     

Hypersurfaces with constant inner curvature of the second fundamental form,and the non-rigidity of the sphere

We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of classC ² where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233, (1972)) saying that such a hypersurface of classC ⁴ has to be a round sphere. In particular, the sphere is notII-rigid in the class of all convexC ²-hypersurfaces.     

Bernstein type result for constant mean curvature hypersurface

We prove a Bernstein type theorem for constant mean curvature hypersurfaces in ℝ ⁿ⁺¹ under certain growth conditions for n ⩽ 3. Our result extends the case when M is a minimal hypersurface in the same condition.      

A Bernstein type theorem in ? × ? n

In this paper, as suitable application of the so-called Omori-Yau generalized maximum principle, we obtain a Bernstein type theorem concerning to complete hypersurfaces immersed with constant mean curvature in the product space ℝ × ℍ ⁿ . Furthermore, we treat the case that such hypersurfaces are vertical graphs.     

SPACELIKE SUBMANIFOLDS IN THE DE SITTER SPACE Sp^（n＋p）（c）WITH CONSTANT SCALAR CURVATURE

Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(n p)(c)， Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of M^n ,respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M^n immersed in Sp^(n p)(c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math, 1998,95 :499-505) is corrected. Moreover,the reduction of the codimension when M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated.     

Hypersurfaces with constant inner curvature of the second fundamental form,and the non-rigidity of the sphere

We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of class C ² where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233, (1972)) saying that such a hypersurface of class C ⁴ has to be a round sphere. In particular, the sphere is not II-rigid in the class of all convex C ² -hypersurfaces. Received 11 October 1994; in final form 26 April 1995     

Generalized bochner theorems and the spectrum of complete manifolds

Bochner's theorem that a compact Riemannian manifold with positive Ricci curvature has vanishing first cohomology group has various extensions to complete noncompact manifolds with Ricci possibly negative. One still has a vanishing theorem for L ² harmonic one-forms if the infimum of the spectrum of the Laplacian on functions is greater than minus the infimum of the Ricci curvature. This result and its analogues for p-forms yield vanishing results for certain infinite volume hyperbolic manifolds. This spectral condition also imposes topological restrictions on the ends of the manifold. More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature; if this average is positive, one has a vanishing theorem for the first cohomology group with compact supports on the universal cover of a compact manifold. There are corresponding results for L ² harmonic spinors on spin manifolds.     

Yamabe Flow and Myers Type Theorem on Complete Manifolds

In this paper, we prove the following Myers type theorem: If (M ⁿ ,g), n≥3, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition Rc≥?Rg, where R>0 is the scalar curvature and ?>0 is a uniform constant, then M ⁿ must be compact.     

The Perron method and the non-linear Plateau problem

We describe a novel technique for solving the Plateau problem for constant curvature hypersurfaces based on recent work of Harvey and Lawson. This is illustrated by an existence theorem for hypersurfaces of constant Gaussian curvature in ${\mathbb{R}^{n+1}}$ .     

On algebraic-geometrical parametrization of the constant mean curvature tori

A theorem providing local coordinates for the constant mean curvature tori in ℝ³ is proved. Related algebraic-geometrical problems arising in analysis of deformations of complex spectra are discussed. Bibliography: 17 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Published inZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 50–64. Translated by R. F. Bikbaev.     

The Minding formula and its applications

We apply the Minding Formula for geodesic curvature and the Gauss-Bonnet Formula to calculate the total Gaussian curvature of certain 2-dimensional open complete branched Riemannian manifolds, the M\cal M surfaces. We prove that for an M\cal M surface, the total curvature depends only on its Euler characteristic and the local behaviour of its metric at ends and branch points. Then we check that many important surfaces, such as complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature, complete constant mean curvature surfaces in hyperbolic 3-space H³ (–1) with finite total curvature, are actually branch point free M\cal M surfaces. Therefore as corollaries we give simple proofs of some classical theorems such as the Chern-Osserman theorem for complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature. For the reader's convenience, we also derive the Minding Formula.     

On an extension of the H k mean curvature flow

In this note,we generalize an extension theorem in [Le-Sesum] and [Xu-Ye-Zhao] of the mean curvature flow to the Hk mean curvature flow under some extra conditions.The main difficulty in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the Hk mean curvature flow,and to do a suitable Moser iteration process.These two problems are overcome by imposing some extra conditions which may be weakened or removed in our forthcoming paper.On the other hand,we derive some estimates for the generalized mean curvature flow,which have their own interesting.