Curvature estimates for properly immersed $\phi_{h}$-bounded submanifolds

TL;DR

This paper extends sharp curvature estimates, both sectional and mean, to a broader class of properly immersed submanifolds called ta-bounded, generalizing previous results for cylindrically bounded cases.

Contribution

It introduces ta-bounded submanifolds and proves sharp curvature estimates for them, broadening the scope of earlier curvature bounds.

Findings

Established sharp sectional curvature estimates for ta-bounded submanifolds.

Proved sharp mean curvature estimates for ta-bounded submanifolds.

Generalized previous results from cylindrically bounded to ta-bounded cases.

Abstract

Jorge-Koutrofiotis and Pigola-Rigoli-Setti proved sharp sectional curvature estimates for extrinsically bounded submanifolds. Alias, Bessa and Montenegro showed that these estimates hold on properly immersed cylindrically bounded submanifolds. On the other hand, Alias, Bessa and Dajczer proved sharp mean curvature estimates for properly immersed cylindrically bounded submanifolds. In this paper we prove these sectional and mean curvature estimates for a larger class of submanifolds, the properly immersed $\phi$-bounded submanifolds.