Energy Decay for the Cauchy Problem of the Linear Wave Equation of Variable Coefficients with Dissipation

Decay of the energy for the Cauchy problem of the wave equation of variable coefficients with a dissipation is considered. It is shown that whether a dissipation can be localized near infinity depends on the curvature properties of a Riemannian metric given by the variable coefficients. In particular, some criteria on curvature of the Riemannian manifold for a dissipation to be localized are given.     

Proper Submanifolds in Product Manifolds

The authors obtain various versions of the Omori-Yau's maximum principle on complete properly immersed submanifolds with controlled mean curvature in certain product manifolds,in complete Riemannian manifolds whose k-Ricci curvature has strong quadratic decay,and also obtain a maximum principle for mean curvature flow of complete manifolds with bounded mean curvature.Using the generalized maximum principle,an estimate on the mean curvature of properly immersed submanifolds with bounded projection in N1 in the product manifold N1 ×N2 is given.Other applications of the generalized maximum principle are also given.     

一类浸入在半黎曼卷积空间的完备超曲面

We deal with complete hypersurfaces immersed in a semi-Riemannian warped product of the type eI×f M~n,where M~n is a connected n-dimensional oriented Riemannian manifold.When the fiber M~n is complete with sectional curvature-k≤K_M for some positive constant k,under appropriate restrictions on the norm of the gradient of the height function h,we proceed with our technique in order to guarantee that complete hypersurface immersed in a semi-Riemannian warped product is a slice.Our approach is based on the well known generalized maximum principle and another suitable maximum principle at the infinity due to Yau.     

HARMONIC FUNCTIONS ON A COMPLETE NONCOMPACT MANIFOLD WITH ASYMPTOTICALLY NONNEGATIVE CURVATURE

The authors prove the space of harmonic functions with polynomial growth of a fixed rate on a complete noncompact Riemannian manifold with asymptotically nonnegative curvature is finite dimensional.     

The α-Geometric Structures on Manifold of Positive Definite Hermite Matrices

Geometric structures of a manifold of positive definite Hermite matrices are considered from the viewpoint of information geometry.A Riemannian metric is defined and dual α-connections are introduced.Then the fact that the manifold is ±l-flat is shown.Moreover,the divergence of two points on the manifold is given through dual potential functions.Furthermore,the optimal approximation of a point onto the submanifold is gotten.Finally,some simulations are given to illustrate our results.     

Liouville Type Theorem About p-Harmonic Function and p-Harmonic Map with Finite L-Energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold (M) of non-negative (n-1)-th Ricci curvature.The Liouville type theorem about the p-harmonic map with finite Lq-energy from complete submanifold in a partially nonnegatively curved manifold to non-positively curved manifold is also obtained.     

Harmonic L~p-functions on Complete Riemannian Manifolds

This survey paper concerns some existence theorems of harmonic functions belonging to LP (M), M being a complete Riemannian manifold. It is well known that a function which is analytic and bounded on the whole complex plane must reduce to a constant.This classical result, known as Liouville's theorem, is also true on a higher-dimensional Euclidean spaces. The generalization of this theorem to other Riemannian manifolds is very interesting. Besides its beauty, the proof usally requires sharp estimates which provide deeper understanding of the Laplacian and hence give broad applications to problems in global analysis.The basic problem in this paper is to study how the geometric conditions of a complete Riemannian manifold affect the validity of the Liouville theorem. The paper consists of two parts. Part I describes the results systematically and Part I will be more technical and will contain the detailed proofs of the results given in the first part.     

Evolution and monotonicity of eigenvalues under the Ricci flow

Let(M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. We derive the evolution equation for the eigenvalues of geometric operator-△φ+ c R under the Ricci flow and the normalized Ricci flow, where △φis the Witten-Laplacian operator, φ∈ C∞(M), and R is the scalar curvature with respect to the metric g(t). As an application, we prove that the eigenvalues of the geometric operator are nondecreasing along the Ricci flow coupled to a heat equation for manifold M with some Ricci curvature condition when c 14.     

MANIFOLD METHOD FOR BOUNDARY VALUE PROBLEM OF TWO ORDER DIFFERENTIAL EQUATIONS

The multiplicity and bifurcation of a two-point boundary value problem(BVP) of nonlinear differential equations(NLDE) are studied firstly using a manifold method.And by some convenient numerical approaches,the discussion on the solutions to the BVP of two order NLDE is turned into the problem on the intersections of the end manifold with the target manifold defined by the boundary conditions,also the bifurcation diagrams and the numerical solutions to the NLDE are obtained.As an example,a mathematical model describing the deformations of elastic circular bar used as gripper and spring element is discussed.     

A Criterion of Nonparabolicity by the Ricci Curvature

A complete manifold is said to be nonparabolic if it does admit a positive Green’s function. To ?nd a sharp geometric criterion for the parabolicity/nonparbolicity is an attractive question inside the function theory on Riemannian manifolds. This paper devotes to proving a criterion for nonparabolicity of a complete manifold weakened by the Ricci curvature. For this purpose, we shall apply the new Laplacian comparison theorem established by the ?rst author to show the existence of a non-constant bounded subharmonic function.     

Flows associated to Cameron-Martin type vector fields on path spaces over a Riemannian manifold

The flow on the Wiener space associated to a tangent process constructed by Cipriano and Cruzeiro, as well as by Gong and Zhang does not allow to recover Driver’s Cameron-Martin theorem on Riemannian path space. The purpose of this work is to refine the method of the modified Picard iteration used in the previous work by Gong and Zhang and to try to recapture and extend the result of Driver. In this paper, we establish the existence and uniqueness of a flow associated to a Cameron-Martin type vector field on the path space over a Riemannian manifold.     

A geometric heat flow for vector fields

We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on K¨ahler manifolds will be studied by Li and Liu(2014).     

关于减小体积的射影映射

Given a projective map F:M→N of a complete Riemannian manifold to a Riemannian manifold with the sectional curvature bounded above by a negative constant,we prove that f decreases volume up to a constant depending only on the curvatures of M and N.This generalizes the result due to Har‘el.     

本刊英文版Vol.27(2011),No.8论文摘要

<正>Schrdinger Soliton from Lorentzian Manifolds Chong SONG You De WANG Abstract In this paper,we introduce a new notion named as Schrdinger soliton.The socalled Schrdinger solitons are a class of solitary wave solutions to the Schrdinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Khler manifold N.If the target manifold N admits a Killing potential,then the Schrdinger soliton reduces to a harmonic     

ON THE INVARIANT SUBMANIFOLDS OF RIEMANNIAN PRODUCT MANIFOLD

In this paper, the vertical and horizontal distributions of an invariant submanifold of a Riemannian product manifold are discussed. An invariant real space form in a Riemannian product manifold is researched. Finally, necessary and sufficient conditions are given on an invariant submanifold of a Riemannian product manifold to be a locally symmetric and real space form.     

An Interpolation of Hardy Inequality and Moser–Trudinger Inequality on Riemannian Manifolds with Negative Curvature

Let M be a complete, simply connected Riemannian manifold with negative curvature.We obtain an interpolation of Hardy inequality and Moser–Trudinger inequality on M. Furthermore,the constant we obtain is sharp.     

Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold Mn × R*

In this paper, the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold Mⁿ × R, where Mⁿ is an n-dimensional (n ≥ 2) complete Riemannian manifold with nonnegative Ricci curvature, and R is the Euclidean 1-space.     

SLANT SUBMANIFOLDS OF A RIEMANNIAN PRODUCT MANIFOLD

In this article, the geometry of the slant submanifolds of a Riemannian product manifold is studied. Some necessary and sufficient conditions on slant, bi-slant and semi-slant submanifolds are given. We research fundamental properties of the distributions which are involved in definitions of semi- and bi-slant submanifolds in a Riemannian product manifold.     

On the hyperbolicity of flows

Let X be a C~1 vector field on a compact boundaryless Riemannian manifold M(dim M≥2),and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = E~sX E~u with E~s uniformly contracting and E~u uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.     

Harnack Differential Inequalities for the Parabolic Equation ut=LF(u) on Riemannian Manifolds and Applications

In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u_t= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.