Curvature estimates for submanifolds immersed into horoballs and horocylinders

We prove mean curvature estimates and a Jorge-Koutroufiotis type theorem for submanifolds confined into either a horocylinder of N X L or a horoball of N, where N is a Cartan-Hadamard manifold with pinched curvature. Thus, these submanifolds behave in many respects like submanifolds immersed into compact balls and into cylinders over compact balls. The proofs rely on the Hessian comparison theorem for the Busemann function.

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

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 $r_varphileq infty$. 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.