Addenda to “The entropy formula for linear heat equation”

Abstract

ABSTRACT. We add two sections to [8] The relation with Li-Yau's gradient estimatesIn this section we provide another derivation of Theorem 1.1 of [8] and discuss its relation with Li-Yau's gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash's 'entropy quantity' − M H log H dv in the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of [9].Let u(x, t) be a positive solution toSimple calculation shows thatMath Subject Classifications. Primary: 58G11.

Entropy and reduced distance for Ricci expanders

“…Our results may be set against recently discovered entropy formulae and pointwise differential inequalities in other settings: For the standard heat equation on a manifold of non negative Ricci curvature [22], and for the forward conjugate heat equation u t = u + Ru on a two-dimensional Ricci flow [23] and (matrix Harnack!) on a Kähler-Ricci flow [24].…”

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

“…1, the entropy formula of type (11) was first obtained by Perelman [33] for the Ricci flow and the conjugate heat equation (2), and was established latter by Ni [29,30] for the linear heat equation ∂ τ u = u on closed Riemannian manifolds. See (3) and (6) respectively.…”

“…See (3) and (6) respectively. In particular, Ni [29,30] proved that W( f, τ ) is monotone decreasing in τ on closed Riemannian manifolds with non-negative Ricci curvature. Using the Li-Yau gradient estimates for the heat kernel of the heat equation ∂ τ u = u, Ni [29,30] …”

“…See also [7,21,27]. Since Perelman's preprint [33] was published on Arxiv in 2002, many people have extended the monotonicity of the W-entropy to other type geometric heat flows on Riemannian manifolds [13,20,25,29,30]. In [29,30], Ni studied the W-entropy for the linear heat equation on complete Riemannian manifolds.…”

“…Inspired by Perlman [33] and Ni [29,30], we establish an analogue of Perelman's and Ni's entropy formula for the W-entropy of the heat equation associated with the Witten Laplacian on complete Riemannian manifolds on which the m-dimensional BakryEmery Ricci curvature is bounded from below and the volume of geodesic balls of a fixed small radius is uniformly bounded from below with their centers. Moreover, we prove a monotonicity theorem and a rigidity theorem for the W-entropy for the Witten Laplacian on complete Riemannian manifolds with non-negative m-dimensional Bakry-Emery Ricci curvature.…”