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

2018/05/25 by Ana Hurtado, Vicente Palmer, Hurtado, Ana +3 · 1 citation
Mathematics · #31C12 #35J25 #53C42 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1805.10055

openalex publication_date 2018/05/25 · openalex created_date 2022/09/20 · openalex updated_date 2026/07/28

Abstract

Let P be a submanifold properly immersed in a rotationally symmetric\nmanifold having a pole and endowed with a weight e^h. The aim of this paper\nis twofold. First, by assuming certain control on the h-mean curvature of\nP, we establish comparisons for the h-capacity of extrinsic balls in P,\nfrom which we deduce criteria ensuring the h-parabolicity or\nh-hyperbolicity of P. Second, we employ functions with geometric meaning to\ndescribe submanifolds of bounded h-mean curvature which are confined into\nsome regions of the ambient manifold. As a consequence, we derive half-space\nand Bernstein-type theorems generalizing previous ones. Our results apply for\nsome relevant h-minimal submanifolds appearing in the singularity theory of\nthe mean curvature flow.\n

Cited by

• Criteria for parabolicity and hyperbolicity of conductive Riemannian manifolds

Related

• A note on the p-Parabolicity of Submanifolds
• Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces
• On p-Parabolicity of Riemannian Submersions
• Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders
• p-Capacity and p-hyperbolicity of submanifolds