Curvature estimates for submanifolds immersed into horoballs and horocylinders

TL;DR

This paper establishes mean curvature bounds and a Jorge-Koutroufiotis type theorem for submanifolds confined within horoballs and horocylinders in Cartan-Hadamard manifolds, extending classical results to non-compact settings.

Contribution

It introduces new curvature estimates and a Jorge-Koutroufiotis type theorem for submanifolds in horoballs and horocylinders, generalizing known results from compact to non-compact ambient spaces.

Findings

Mean curvature estimates for submanifolds in horoballs and horocylinders.

A Jorge-Koutroufiotis type theorem applicable to these submanifolds.

Submanifolds behave similarly to those in compact balls in terms of curvature properties.

Abstract

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.