An estimate for the sectional curvature of cylindrically bounded submanifolds

TL;DR

This paper provides precise estimates for the sectional curvature of certain complete submanifolds in product spaces, extending classical results under new growth and decay conditions.

Contribution

It introduces sharp curvature bounds for cylindrically bounded submanifolds, extending the Jorge-Koutrofiotis Theorem with new growth and decay assumptions.

Findings

Sharp curvature estimates for cylindrically bounded submanifolds.

Extension of the Jorge-Koutrofiotis Theorem.

Conditions under which curvature bounds hold.

Abstract

We give sharp sectional curvature estimates for complete immersed cylindrically bounded $m$-submanifolds $\phi:M\to N\times\mathbb{R}^{\ell}$, $n+\ell\leq 2m-1$ provided that either $\phi$ is proper with the second fundamental form with certain controlled growth or $M$ has scalar curvature with strong quadratic decay. This latter gives a non-trivial extension of the Jorge-Koutrofiotis Theorem [7]