Relating diameter and mean curvature for Riemannian submanifolds

TL;DR

This paper establishes a relationship between the diameter and mean curvature integral of Riemannian submanifolds, extending previous Euclidean results to more general ambient manifolds.

Contribution

It generalizes Topping's Euclidean diameter estimate to submanifolds immersed in arbitrary Riemannian manifolds under certain geometric conditions.

Findings

Diameter bounds in terms of mean curvature integral

Extension of Euclidean results to general Riemannian ambient spaces

Provides new geometric inequalities for submanifolds

Abstract

Given an $m$-dimensional closed connected Riemannian manifold $M$ smoothly isometrically immersed in an $n$-dimensional Riemannian manifold $N$, we estimate the diameter of $M$ in terms of its mean curvature field integral under some geometric restrictions, and therefore generalize a recent work of Topping in the Euclidean case (Comment. Math. Helv., 83 (2008), 539--546).