A BRAY-BRENDLE-NEVES TYPE INEQUALITY FOR A RIEMANNIAN MANIFOLD

In this paper,for any local area-minimizing closed hypersurface∑with RcΣ=RΣ/ngΣ,immersed in a(n+1)-dimension Riemannian manifold M which has positive scalar curvature and nonnegative Ricci curvature,we obtain an upper bound for the area of∑.In particular,when∑saturates the corresponding upper bound,∑is isometric to S^(n)and M splits in a neighborhood of∑.At the end of the paper,we also give the global version of this result.

GEOMETRIC INEQUALITIES FOR CERTAIN SUBMANIFOLDS IN A PINCHED RIEMANNIAN MANIFOLD

这篇文章必要时为与平行第二基本形式歧管的一个代用品给一些几何不平等歧管的 Riemannian 和为它的第二个基础的方形的标准的分发形成。给词调音：Submanifolds；第二种基本形式；拧的 Riemannian

The First Initial Boundary Value Problem for Parabolic Hessian Equations on Riemannian Manifolds

For a class of elliptic Hessian operators, one type of corresponding parabolic Hessian equations is studied on Riemannian manifolds. The existence and uniqueness of the admissible solution to the frst initial boundary value problem for the equations are shown.

Lower Bound of the First Eigenvalue for the Laplace Operator on Compact Riemannian Manifold

Let M be a compact m-dimensional Riemannian manifold,let d denote its diameter,-R(R>0) the lower bound of the Ricci curvature,and λ1 the first eigenvalue for the Laplacian on M.Then there ex-ists a constant Cm=max｛√m-1,√2｝,such that λ1≥π^2/d^2·1/(2-11/2π^2)+11/2π^2e^Cm√Rd^2.

An Estimate on Riemannian Manifolds of Dimension 4

We give an estimate of type sup × inf on Riemannian manifold of dimension 4 for a Yamabe type equation.

The Harmonic Functions on a Complete Asymptotic Flat Riemannian Manifold

Let be a simply connected complete Riemannian manifold with dimension n≥3 . Suppose that the sectional curvature satisfies , where p is distance function from a base point of M, a, b are constants and . Then there exist harmonic functions on M .

Well-Posedness of Stochastic Continuity Equations on Riemannian Manifolds

The authors analyze continuity equations with Stratonovich stochasticity,■ρ+divh[ρo(u(t,x)+∑_(i=1)^(N)a_(i)(x)w_(i)(t))]=0defined on a smooth closed Riemannian manifold M with metric h.The velocity field u is perturbed by Gaussian noise terms Wi(t),:WN(t)driven by smooth spatially dependent vector fields a1(x),...,aN(x)on M.The velocity u belongs to L_(t)^(1)W_(x)^(1,2)with divh u bounded in Lf,for p>d+2,where d is the dimension of M(they do not assume div_(h) u∈L_(t,x)^(∞)).For carefully chosen noise vector fields ai(and the number N of them),they show that the initial-value problem is well-posed in the class of weak L^(2) solutions,although the problem can be ill-posed in the deterministic case because of concentration effects.The proof of this“regularization by noise”result is based on a L^(2) estimate,which is obtained by a duality method,and a weak compactness argument.

A Discussion on the Establishment That a Fibre Metric on the Positive Definite Real Inner-Product of a Properly Embedded Smooth Submanifold Be Always Extended to a Riemannian Metric on the Positive Definite Real Inner-Product

Let M be a smooth manifold and S ⊆ M a properly embedded smooth submanifold. Suppose that we have a fibre metric on TM|s i.e. a positive definite real inner-product on TpM for all p ∈ S, which depends smoothly on p ∈ S. The purpose of this article is to figure out that the fibre metric on TM|s can always be extended to a Riemannian metric on TM from a special perspective.

On the p-Laplacian Lichnerowicz equation on compact Riemannian manifolds

In this paper,we deal with a singular quasilinear critical elliptic equation of Lichnerowicz type involving the p-Laplacian operator.With the help of the subcritical approach from the variational method,we obtain the non-existence,existence,and multiplicity results under some given assumptions.

Nonparametric estimation for stationary and strongly mixing processes on Riemannian manifolds

In this paper,nonparametric estimation for a stationary strongly mixing and manifoldvalued process(X_(j))is considered.In this non-Euclidean and not necessarily i.i.d setting,we propose kernel density estimators of the joint probability density function,of the conditional probability density functions and of the conditional expectations of functionals of X_(j)given the past behavior of the process.We prove the strong consistency of these estimators under sufficient conditions,and we illustrate their performance through simulation studies and real data analysis.

General formula for lower bound of the first eigenvalue on Riemannian manifolds

A general formula for the lower bound of the first eigenvalue on compact Riemannian manifolds is presented. The formula improves the main known sharp estimates including Lichnerowicz’s estimate and Zhong-Yang’s estimate. Moreover, the results are extended to the noncompact manifolds. The study is based on the probabilistic approach (i.e. the coupling method).

The extension for mean curvature flow with finite integral curvature in Riemannian manifolds

We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature at a finite time interval [0,T) can be extended over time T. Moreover,we show that the condition is optimal in some sense.

DARBOUX EQUATIONS AND ISOMETRIC EMBEDDING OF RIEMANNIAN MANIFOLDS WITH NONNEGATIVE CURVATURE IN R

The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate onthe boundary is contained As some applications the smooth isometric embeddings of positivelyand nonnegatively curved disks into R^3 are constructed.

A class of Kazdan-Warner typed equations on non-compact Riemannian manifolds

In this paper,we discuss a Kazdan-Warner typed equation on certain non-compact Rie- mannian manifolds.As an application,we prove an existence theorem of Hermitian-Yang-Mills-Higgs metrics on holomorphic line bundles over certain non-compact K(?)hler manifolds.

Every Sub-Riemannian Manifold Is the Gromov–Hausdorff Limit of a Sequence Riemannian Manifolds

In this paper, we will show that every sub-Riemannian manifold is the Gromov–Hausdorff limit of a sequence of Riemannian manifolds.

Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures

By establishing the intrinsic super-Poincar'e inequality,some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive.These conditions,as well as the resulting uniform upper bounds on the intrinsic heat kernels,are sharp for some concrete examples.

Convergence of the Newton method and uniqueness of zeros of vector fields on Riemannian manifolds

The estimates of the radii of convergence balls of the Newton method and uniqueness balls of zeroes of vector fields on the Riemannian manifolds are given under the assumption that the covariant derivatives of the vector fields satisfy some kind of general Lipschitz conditions. Some classical results such as the Kantorovich's type theorem and the Smale's γ-theory are extended.

Nonlinear Damped Oscillators on Riemannian Manifolds:Fundamentals

The classical theory of mass-spring-damper-type dynamical systems on the ordinary flat space R^3 may be generalized to higher-dimensional Riemannian manifolds by reformulating the basic underlying physical principles through differential geometry.Nonlinear dynamical systems have been studied in the scientific literature because they arise naturally from the modeling of complex physical structures and because such dynamical systems constitute the basis for several modern applications such as the secure transmission of information.The flows of nonlinear dynamical systems may evolve over time in complex,non-repeating(although deterministic) patterns.The focus of the present paper is on formulating the general equations that describe the dynamics of a point-wise particle sliding on a Riemannian manifold in a coordinate-free manner.The paper shows how the equations particularize in the case of some manifolds of interest in the scientific literature,such as the Stiefel manifold and the manifold of symmetric positive-definite matrices.

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.

Embeddings of Almost Hermitian Manifold in Almost Hyper Hermitian Manifold and Complex (Hypercomplex) Numbers in Riemannian Geometry

Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold. Further, an almost hyper Hermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the tangent bundle TM equipped with the deformed almost hyper Hermitian structure of the special form. As a result, we have obtained that any Riemannian manifold M of dimension n can be embedded as a totally geodesic submanifold in a Kaehlerian manifold of dimension 2n (Theorem 6) and in a hyper Kaehlerian manifold of dimension 4n (Theorem 7). Such embeddings are “good” from the point of view of Riemannian geometry. They allow solving problems of Riemannian geometry by methods of Kaehlerian geometry (see Section 5 as an example). We can find similar situation in mathematical analysis (real and complex).