# Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

## TL;DR

This paper establishes comparison results for isoperimetric quotients and capacities in weighted manifolds under curvature bounds, leading to criteria for parabolicity and hyperbolicity, and extends analysis to submanifolds with controlled weighted mean curvature.

## Contribution

It provides new comparison theorems for weighted isoperimetric quotients and capacities in weighted manifolds with curvature bounds, extending previous results and including submanifold analysis.

## Key findings

- Comparison results for weighted isoperimetric quotients
- Criteria for parabolicity and hyperbolicity in weighted manifolds
- Extension of techniques to submanifolds with controlled weighted mean curvature

## Abstract

Let $(M,g)$ be a complete non-compact Riemannian manifold together with a function $e^h$, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of $M$ to deduce comparisons for the weighted isoperimetric quotient and the weighted capacity of metric balls in $M$ centered at a point $o\in M$. As a consequence, we obtain parabolicity and hyperbolicity criteria for weighted manifolds generalizing previous ones. A basic tool in our study is the analysis of the weighted Laplacian of the distance function from $o$. The technique extends to non-compact submanifolds properly immersed in $M$ under certain control on their weighted mean curvature.