GEOMETRIC INEQUALITIES FOR CERTAIN SUBMANIFOLDS IN A PINCHED RIEMANNIAN MANIFOLD

This article gives some geometric inequalities for a submanifold with parallel second fundamental form in a pinched Riemannian manifold and the distribution for the square norm of its second fundamental form.

Complex Networks from Chaotic Time Series on Riemannian Manifold

Complex networks are important paradigms for analyzing the complex systems as they allow understanding the structural properties of systems composed of different interacting entities. In this work we propose a reliable method for constructing complex networks from chaotic time series. We first estimate the covariance matrices, then a geodesic-based distance between the covariance matrices is introduced. Consequently the network can be constructed on a Riemannian manifold where the nodes and edges correspond to the covariance matrix and geodesic-based distance, respectively. The proposed method provides us with an intrinsic geometry viewpoint to understand the time series.

An Estimate on Riemannian Manifolds of Dimension 4

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

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.

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. uniqueness of the admissible solution to the first initial boundary the equations are shown. The existence and value problem for

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.

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.

Multiple Solutions for an Elliptic Equation with Hardy Potential and Critical Nonlinearity on Compact Riemannian Manifolds

We prove the existence of multiple solutions of an elliptic equation with critical Sobolev growth and critical Hardy potential on compact Riemannian manifolds.

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.

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.

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.

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.

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.

DIFFUSION PROCESSES ON COMPLETE RIEMANNIAN MANIFOLDS

In this paper,a basic estimate for the conditional Riemannian Brownian motion on a complete manifold with non-negative Ricci curvature is established.Applying it to the heat kernel estimate of the operator 1/2△+b,we obtain the Aronson′s estimate for the operator 1/2△+b,which can be regarded as an extension of Peter Li and S.T.Yau's heat kernel estimate for the Laplace-Beltrami operator.

Cohomogeneity Two Actions on Flat Riemannian Manifolds

In this paper, we study flat Riemannian manifolds which have codimension two orbits, under the action of a closed and connected Lie group G of isometries. We assume that G has fixed points, then characterize M and orbits of M.

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.