Panagiotis Gianniotis, Department of Pure Mathematics, University of Waterloo
"Relating diameter and mean curvature for submanifolds of Euclidean space"
We will talk about an estimate of Peter Topping in relation to the intrinsic geometry of closed Riemannian manifolds that are isometrically immersed in Euclidean space. In particular, Topping bounds the (intrinsic) diameter of the immersed submanifold solely in terms of the mean curvature of the immersion. The result improves a previous estimate by Leon Simon, and uses a Sobolev inequality due to Michael and Simon, valid for any immersion in Euclidean space. Topping shows how to relate a Sobolev inequality to diameter, a technique that he also applies to the Ricci flow to obtain a bound on the diameter of the evolving manifold in terms of its scalar curvature.
MC 5479