First Variation of the Log Entropy Functional along the Ricci Flow

In this note, we establish the first variation formula of the adjusted log entropy functional Ya introduced by Ye in [14]. As a direct consequence, we also obtain the monotonicity of Ya along the Ricci flow. Various entropy functionals play crucial role in the singularity analysis of Ricci flow. Let (M, g(t)) be a smooth family of Riemannian metrics on a closed manifold M and suppose g(t) is a solution of Hamilton’s Ricci flow equation. In a recent interesting paper [14], R. Ye introduced a new entropy functional, the adjusted log entropy, as follows (1.1) Ya(g, u, t) = − ∫ M u lnudvol + n 2 ln ( ∫ M (|∇u| + R 4 u)dvol + a ) + 4at, where the positive function u ∈ W 1,2(Mn) satisfies ∫ M (|∇u|2 + R4 u 2)dvol + a > 0, and R denotes the scalar curvature of the metric at time t. The log entropy functional can be used to prove uniform logarithmic Sobolev inequalities along the Ricci flow which also leads to uniform Sobolev inequalities, see Ye’s recent series of papers, [13], [14], etc, and Zhang [15]. This new entropy functional of Ye shares a similar important feature with Perelman’s entropy functionals. Namely, it is nondecreasing under the following coupled system of Ricci flow,