Diameter estimates for submanifolds in manifolds with nonnegative curvature

TL;DR

This paper provides new estimates for the intrinsic diameter of submanifolds immersed in manifolds with nonnegative curvature, relating it to their mean curvature and boundary length, extending previous results in geometric analysis.

Contribution

It introduces diameter bounds for immersed submanifolds based on mean curvature integrals, generalizing earlier work to broader classes of manifolds and submanifold types.

Findings

Diameter bounds in terms of mean curvature integral

Extension of previous diameter estimates

Applicability to convex surfaces with boundary

Abstract

Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of Topping, Wu-Zheng and Paeng.