Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights. (March 2020)

• Record Type: Journal Article
• Authors: Hurtado, A.; Palmer, V.; Rosales, C.
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: 31C12 -- 53C42 -- 35J25; Rotationally symmetric spaces -- Weights -- Submanifolds -- Laplacian -- Parabolicity -- Capacity -- Mean curvature -- Half-space theorem -- Bernstein theorem; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111681
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: 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 and Bernstein-type theorems generalizing previous ones. Our results apply for some relevant h -minimal submanifolds appearing in the singularity theory of the mean curvature flow.