Let M be a complete Riemannian manifold possibly with a boundary ∂M. For any C1-vector field Z, by using gradient/functional inequalities of the (reflecting) diffusion process generated by L:= Δ+Z, pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of ∂M if it exists. These characterizations extend and strengthen the recent results derived by Naber for the uniform norm ||RicZ||∞ on manifolds without boundaries. A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author, such that the proofs are significantly simplified.
curvature second fundamental form diffusion process path space
This is a preview of subscription content, log in to check access.
Capitaine B, Hsu E P, Ledoux M. Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron Comm Probab, 1997, 2: 71–81MathSciNetCrossRefMATHGoogle Scholar