Relating diameter and mean curvature for Riemannian submanifolds

Abstract

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