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

Abstract

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

13 pages

References in corpus (1)

• The mean curvature of cylindrically bounded submanifolds