Spectral, stochastic and curvature estimates for submanifolds of highly negative curved spaces

TL;DR

This paper establishes spectral, stochastic, and curvature estimates for complete submanifolds in negatively curved spaces, linking geometric properties to isoperimetric ratios and extrinsic radius, applicable to both bounded and unbounded cases.

Contribution

It provides new estimates for submanifolds in negatively curved spaces, extending results to unbounded cases with conditions on isoperimetric ratios and curvature.

Findings

Estimates hold for both bounded and unbounded extrinsic radius cases.

Integrability of inverse isoperimetric ratio is key for unbounded case.

Highly negative curvature ensures conditions for unbounded case.

Abstract

We prove spectral, stochastic and mean curvature estimates for complete $m$-submanifolds $\varphi \colon M \to N$ of $n$-manifolds with a pole $N$ in terms of the comparison isoperimetric ratio $I_{m}$ and the extrinsic radius $r_\varphi\leq \infty$. Our proof holds for the bounded case $r_\varphi< \infty$, recovering the known results, as well as for the unbounded case $r_{\varphi}=\infty$. In both cases, the fundamental ingredient in these estimates is the integrability over $(0, r_\varphi)$ of the inverse $I_{m}^{-1}$ of the comparison isoperimetric radius. When $r_{\varphi}=\infty$, this condition is guaranteed if $N$ is highly negatively curved.