Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below

TL;DR

This paper establishes bounds on isoperimetric quotients and mean exit times for submanifolds with curvature bounds, providing conditions for parabolicity and extending geometric analysis techniques to new curvature regimes.

Contribution

It introduces new upper and lower bounds for isoperimetric quotients and mean exit times under lower curvature bounds, complementing previous upper-bound results.

Findings

Derived upper bounds for isoperimetric quotients of extrinsic balls.

Established lower bounds on mean exit time for Brownian motion.

Provided conditions for submanifolds to be parabolic based on capacity comparison.

Abstract

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motion in the extrinsic balls. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above.