Volume growth and holonomy in nonnegative curvature

The volume growth of an open manifold of nonnegative sectional curvature is proven to be bounded above by the difference between the codimension of the soul and the maximal dimension of an orbit of the action of the normal holonomy group of the soul. Additionally, an example of a simply-connected soul with a non-compact normal holonomy group is constructed.

     

Green's function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature

We extend the method of harmonic transplantation from Euclidean domains to spaces of constant positive or negative curvature. To this end the structure of the Green's function of the corresponding Laplace-Beltrami operator is investigated. By means of isoperimetric inequalities we derive complementary estimates for its distribution function. We apply the method of harmonic transplantation to the question of whether the best Sobolev constant for the critical exponent is attained, i.e. whether there is an extremal function for the best Sobolev constant in spaces of constant curvature. A fairly complete answer is given, based on a concentration-compactness argument and a Pohozaev identity. The result depends on the curvature.

     

Volume growth and heat kernel estimates for the continuum random tree

In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost–surely logarithmic global fluctuations and log–logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r → 0. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t → 0 almost–surely. Finally, we prove that this quenched (almost–sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth.      

Closed geodesics and volume growth of open manifolds with sectional curvature bounded from below

In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.     

非负曲率流形的体积增长估计

本文利用非负曲率流形上的Busemann函数和穷竭函数的性质,得出了在某紧致子集外满足一定非负曲率条件的完备非紧的(复) n维K■hler流形的体积增长至少是n次的．推广了陈兵龙和朱熹平教授新近的一个结果．     

Cauchy problem for \left\{ {\vec p;\vec h} \right\}-parabolic equations with time-dependent coefficients

We establish the existence of a unique solution continuously depending on the initial data to the Cauchy problem for -parabolic equations with time-dependent coefficients for which the initial data are generalized functions (distributions) of slow growth. For a particular class of equations, we state necessary and sufficient conditions for the existence of a unique solution of the Cauchy problem with properties of its spatial variable which are characteristic of its fundamental solution.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 395–411.Original Russian Text Copyright © 2005 by V. A. Litovchenko.This revised version was published online in April 2005 with a corrected issue number.     

An estimate for the three-dimensional discrete Green's function and applications

In this paper we first introduce definitions of the regularized Green's function and the discrete Green's function in three dimensions. With complicated arguments estimates for these two functions are then derived. Finally we give an application of the estimate for the discrete Green's function to the high accuracy analysis of the three-dimensional block finite element approximation.     

Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth

In this note we construct two quasi-isometric graphs. One admits an infinite dimensional space of nonconstant bounded harmonic functions, while the other admits only constant bounded harmonic functions. Translation of the construction to manifolds answers a problem due to T. Lyons.     

Infinity Behavior of Bounded Subharmonic Functions on Ricci Non-negative Manifolds

In this paper,we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M.We first show that limr→∞r^2/V(r)∫B(r)△hdv=0if h is a bounded subharmonic function.If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity,under certain auxiliary conditions on the volume growth of M.In particular,our result applies to the case when the Riemannian manifold has maximum volume growth.We also derive a representation formula in our paper,from which one can easily derive Yau‘s Liouville theorem on bounded harmonic functions.     

Planar Riemann surfaces with uniformly distributed cusps: parabolicity and hyperbolicity

We consider a planar Riemann surface R made of a non‐compact simply connected plane domain from which an infinite discrete set of points is removed. We give several conditions for the collars of the cusps in R caused by these points to be uniformly distributed in R in terms of Euclidean geometry. Then we associate a graph G with R by taking the Voronoi diagram for the uniformly distributed cusps and show that G represents certain geometric and analytic properties of R .     

锥中一类调和函数的增长估计

给出了锥中一类调和函数在无穷远点处的增长估计,推广了Siegel和Talvila,张和邓在半空间的相关结果.     

Volume entropy based on integral Ricci curvature and volume ratio

For a compact Riemannian manifold M, we obtain an explicit upper bound of the volume entropy with an integral of Ricci curvature on M and a volume ratio between two balls in the universal covering space.     

Excessiveness of harmonic functions for certain diffusions

In an earlier paper⁽⁴⁾ the author has shown that a diffusion process whose potential kernel satisfies certain analytic conditions has all of its excessive harmonic functions, which are not identically infinite, continuous. This paper shows that under these conditions (concerning its potential kernel), the excessiveness of its nonnegative harmonic functions isautomatic.     

Volume growth and escape rate of symmetric diffusion processes

We give an upper rate function, in terms of the volume growth of the underlying state space, for the symmetric diffusion process associated with a symmetric, strongly local regular Dirichlet form. It extends the main result of Hsu and Qin [Ann. Probab. 38(4) 2010], where an upper rate function was given for Brownian motion on Riemannian manifold.     

Volume, surface area and inward injectivity radius

We show some relations among the volumes of domains in Euclidean spaces, their surface areas and the inward injectivity radii from their boundaries. In particular, we give an estimate for the upper bound of the ratios of their surface areas and volumes by means of inward injectivity radii. The upper bound seems to depend on their topological structures.

     

Volume growth for gradient shrinking solitons of Ricci-harmonic flow

In this paper,we derive an estimate on the potential functions of complete noncompact gradient shrinking solitons of Ricci-harmonic flow,and show that complete noncompact gradient shrinking Ricci-harmonic solitons have Euclidean volume growth at most.     

Volume and distance comparison theorems for sub-Riemannian manifolds

In this paper we study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced by the first and the third author in [3] and its use to obtain sharp inequalities for solutions of the sub-Riemannian heat equation. As a consequence, we obtain a Gromov type precompactness theorem for the class of sub-Riemannian manifolds whose generalized Ricci curvature is bounded from below in the sense of [3].     

具有非负Ricci曲率和大体积增长的开流形

In this paper,we prove that a complete n-dimensional Riemannian manifold with n0nnegative kth-Ricci curvature,large volume growth has finite topological type provided that lim{((vol[B(p,r))]/(ω_nr~n)-αM)r(k(n-1))/(k 1)(1-α/2)}＜=εfor some constantε＞0.We also prove that a complete Riemannian manifold with nonnegative kth-Ricci curvature and under some pinching conditions is diffeomorphic to R~n.     

Numerical differentiation for two-dimensional scattered data

In this paper, we propose a regularization method for numerical differentiation of two-dimensional mildly scattered input data. A regularized solution is constructed based on the Green's function. The existence and uniqueness of the regularized solution are proved and the convergence estimates are provided under a simple choice of regularization parameter. Numerical results show that our method is quite effective. One of the advantages for our proposed method is that the basis functions are independent of input data.     

Strongly regular growth of entire functions of order zero

For entire functions of order zero we introduce a new concept of regularity of growth, which is shown to possess properties similar to those which characterize the concept of totally regular growth of entire functions of finite order in the sense of Levin-Pflüger. Translated fromMatematicheskie Zameiki, Vol. 63, No. 2, pp. 196–208, February, 1998. This research was partially supported by the International Science Foundation under grant No. UCR000.