Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain

In this paper we introduce a weighted Cheeger constant and show that the gap between the first two eigenvalues of a Riemannian manifold given Dirichlet conditions can be bounded from below in terms of this constant. When the Riemannian manifold is a bounded Euclidean domain satisfying an interior rolling sphere condition we give an estimate on the weighted Cheeger constant in terms of the rolling sphere radius, volume, a bound on the principal curvatures of the boundary and the dimension. This yields a lower bound on the nontrivial gap for Euclidean domains. S-Y. Cheng’s research partially supported by the CUHK direct grant A/C # 220600260. K. Oden’s research partially supported by the Department of Education Graduate Fellowship     

Rigidity and sphere theorem for manifolds with positive Ricci curvature

LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound. Supported by the JSPS postdoctoral fellowship and NSF of China     

Restrictions on collapsing with a lower sectional curvature bound

We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with sectional curvature bounded from below.     

n-Marked黎曼球面的对称性

本文讨论了n-marked黎曼球面模空间上自然的对称群作用,计算出其全部的不动点及n-marked黎曼球面的对称类型,同时讨论了黎曼球面上SL(2,C)作用的动力系统特征,得出其周期性的一个描述,并在这些基础上给出n-punctured黎曼球面模空间的奇性刻画．     

Quantitative Isoperimetry à la Levy-Gromov

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the manifold and of the corresponding sphere is likewise bounded. These results are actually obtained in the more general context of (possibly nonsmooth) metric measure spaces with curvature-dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method. © 2018 Wiley Periodicals, Inc.     

Eigenvalue problems on Riemannian manifolds with a modified Ricci tensor

In this paper, we study the eigenvalue problems on a Riemannian manifold with a modified Ricci tensor. We obtain some sharp lower bound estimates for the first eigenvalue of Laplacian. We also prove some rigidity theorems for the Riemannian manifold with some suitable conditions.     

Improved sphere packing lower bounds from Hurwitz lattices

In this paper we prove an asymptotic lower bound for the sphere packing density in dimensions divisible by four. This asymptotic lower bound improves on previous asymptotic bounds by a constant factor and improves not just lower bounds for the sphere packing density, but also for the lattice sphere packing density and, in fact, the Hurwitz lattice sphere packing density.     

Critical points of the total scalar curvature functional on the space of metrics of constant scalar curvature

It is well known that critical points of the total scalar curvature functional ? on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of ? is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also Einstein or isometric to a standard sphere. In this paper we prove that n-dimensional critical points have vanishing n− 1 homology under a lower Ricci curvature bound for dimension less than 8. Received: 12 July 1999     

A Poincaré-Sobolev type inequality on compact Riemannian manifolds with boundary

In this paper we study a Poincaré-Sobolev type inequality on compact Riemannian n-manifolds with boundary where the exponent growth is critical. Two constants have to be determined. We show that, contrary to the classical Sobolev inequality, the first best constant in this inequality does not depend on the dimension only, but depends on the geometry. It can be represented as the minimum of a given energy functional. We study the nonlinear PDE associated to this functional which involves the geometry of the boundary. For a star-shaped domain D in whose boundary has positive Ricci curvature, we give explicitly two Sobolev constants corresponding to the embedding in . This result is used to obtain an explicit geometrical lower bound for . Received November 15, 1999 / Published online April 12, 2001     

Closed manifolds with small excess

In this article, we study closed Riemannian manifolds with small excess. We show that a closed connected Riemannian manifold with Ricci curvature and injectivity radius bounded from below is homeomorphic to a sphere if it has sufficiently small excess. We also show that a closed connected Riemannian manifold with weakly bounded geometry is a homotopy sphere if its excess is small enough.     

Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem

In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume.The related questions that will also be studied are the following: given a contractible k-dimensional sphere in M ⁿ , how “fast” can this sphere be contracted to a point, if π _i (M ⁿ )={0} for 1≤i<k. That is, what is the maximal length of the trajectory described by a point of a sphere under an “optimal” homotopy? Also, what is the “size” of the smallest non-contractible k-dimensional sphere in a (k-1)-connected manifold M ⁿ providing that M ⁿ is not k-connected?     

A generalization of the classical sphere theorem

In this paper, we prove a sphere theorem for Riemannian manifolds with partially positive curvature which generalizes the classical sphere theorem.

     

Riemann流形上Laplace算子第一特征值的一点注记

本文研究了黎曼流形上Laplace算子的第一特征值,利用流形的测地球上的Sobolev常数进行讨论并进行Moser迭代,得到闭的黎曼流形上Laplace算子第一特征值的一个下界估计.     

A differentiable sphere theorem on manifolds with reverse volume pinching

Based on the celebrated 1/4-pinching sphere theorem, we prove a differentiable sphere theorem on Riemannian manifolds with reverse volume pinching.     

Reilly-type inequalities for p-Laplacian on compact Riemannian manifolds

For a compact Riemannian manifold M immersed into a higher dimensional manifold which can be chosen to be a Euclidean space, a unit sphere, or even a projective space, we successfully give several upper bounds in terms of the norm of the mean curvature vector of M for the first non-zero eigenvalue of the p-Laplacian (1<p<+∞) on M. This result can be seen as an extension of Reilly’s bound for the first non-zero closed eigenvalue of the Laplace operator.     

Volume collapsed three-manifolds with a lower curvature bound

In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.     

Ricci curvature,radial curvature and large volume growth

In this paper, we study complete noncompact Riemannian manifolds with Ricci curvature bounded from below. When the Ricci curvature is nonnegative, we show that this kind of manifolds are diffeomorphic to a Euclidean space, by assuming an upper bound on the radial curvature and a volume growth condition of their geodesic balls. When the Ricci curvature only has a lower bound, we also prove that such a manifold is diffeomorphic to a Euclidean space if the radial curvature is bounded from below. Moreover, by assuming different conditions and applying different methods, we shall prove more results on Riemannian manifolds with large volume growth.     

The spectral sequence of the canonical foliation of a Vaisman manifold

In this paper we investigate the spectral sequence associated with a Riemannian foliation which arises naturally on a Vaisman manifold. Using the Betti numbers of the underlying manifold we establish a lower bound for the dimension of some terms of this cohomological object. This way we obtain cohomological obstructions for two-dimensional foliations to be induced from a Vaisman structure. We show that if the foliation is quasi-regular the lower bound is realized. In the final part of the paper we discuss two examples.     

Geodesic vector fields and Eikonal equation on a Riemannian manifold

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.