05e2a0647e260c355dd2b2175edb45b8-Supplemental.pdf

One classical example is that the Riemannian manifoldEm = (Rm,h,iRm) is nothing but the m-dimensionalEuclideanspace. Assume that the heat kernel is Lipschitz continous. We first construct a sequence of probability measures{ρt}t N, such that ρ2t = µtφφφ, ρ2t+1 = µtM, t N. Proof For a given manifoldM, its Riemannian volume element doesn't vary at different time t, thus thelowerbound ofintegral R