Uniform Lipschitz continuity of the isoperimetric profile of compact surfaces under normalized Ricci flow

Abstract

We show that the isoperimetric profile h g ( t ) ( ξ ) h_{g(t)}(\xi ) of a compact Riemannian manifold ( M , g ) (M,g) is jointly continuous when metrics g ( t ) g(t) vary continuously. We also show that, when M M is a compact surface and g ( t ) g(t) evolves under normalized Ricci flow, h g ( t ) 2 ( ξ ) h^2_{g(t)}(\xi ) is uniform Lipschitz continuous and hence h g ( t ) ( ξ ) h_{g(t)}(\xi ) is uniform locally Lipschitz continuous.