On the characterization of parabolicity and hyperbolicity of submanifolds

Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights

Let $P$ be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight $e^h$. The aim of this paper is twofold. First, by assuming certain control on the $h$-mean curvature of $P$, we establish comparisons for the $h$-capacity of extrinsic balls in $P$, from which we deduce criteria ensuring the $h$-parabolicity or $h$-hyperbolicity of $P$. Second, we employ functions with geometric meaning to describe submanifolds of bounded $h$-mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space a…

A note on the p-Parabolicity of Submanifolds

We give a geometric criterion which shows p-parabolicity of a class of submanifolds in a Riemannian manifold, with controlled second fundamental form, for p bigger or equal than 2.

p-Capacity and p-hyperbolicity of submanifolds

We use drifted Brownian motion in warped product model spaces as comparison constructions to show $p$-hyperbolicity of a large class of submanifolds for $p\ge 2$. The condition for $p$-hyperbolicity is expressed in terms of upper support functions for the radial sectional curvatures of the ambient space and for the radial convexity of the submanifold. In the process of showing $p$-hyperbolicity we also obtain explicit lower bounds on the $p$-capacity of finite annular domains of the submanifolds in terms of the drifted 2-capacity of the corresponding annuli in the respective comparison spaces.

导电黎曼流形的抛物性与双曲性判据

Criteria for parabolicity and hyperbolicity of conductive Riemannian manifolds

本文研究各向异性导电黎曼流形的分类问题，利用度量与自导电导率张量建立 $\mathcal{W}$-抛物性和 $\mathcal{W}$-双曲性的内在与外在判据，并通过推广及新构造的例子验证其结论。

Parabolicity, Brownian escape rate and properness of self-similar solutions of the direct and inverse Mean Curvature Flow

We study some potential theoretic properties of homothetic solitons $Σ^n$ of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in $\mathbb{R}^{n+m}$, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons $Σ^n$ with $n>2$ are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the Mean Exit Time function for the Brow…

关于伪双曲空间中完备极大类空子流形的标量曲率

On the scalar curvature of complete maximal spacelike submanifolds in pseudo-hypebolic spaces

本文研究了伪双曲空间中完备极大类空子流形的曲率，证明了其标量曲率在任意符号下均非正并给出精确界限，显式刻画了达到界限的子流形，并在余维数为1时推导出Ricci曲率的精确上界。