Gradient estimate for the forward conjugate heat equation of forms under the Ricci flow

We establish bounds for the gradient of solutions to the forward conjugate heat equation of differential forms on a Riemannian manifold with the metric evolves under the Ricci flow.

Heat Kernel Estimates Under the Ricci–Harmonic Map Flow

This paper considers the Ricci flow coupled with the harmonic map flow between two manifolds. We derive estimates for the fundamental solution of the corresponding conjugate heat equation and we prove an analogue of Perelman's differential Harnack inequality. As an application, we find a connection between the entropy functional and the best constant in the Sobolev embedding theorem in ℝn.

Differential Harnack estimates for conjugate heat equation under the Ricci flow

We prove (local and global) differential Harnack inequalities for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace–Beltrami operator is replaced with “[Formula: see text]”, where [Formula: see text] is the scalar curvature of the Ricci flow, which is well generalized to the case of nonlinear heat equation with potential. Our estimates improve on some well known results by weakening the curvature constraints. As a by-product, we obtain some Li–Yau-type differential Harnack estimate. The localized version of our estimate is very useful in extending the results obtained to noncompact case.

Pseudolocality for the Ricci Flow and Applications

Perelman established a differential Li-Yau-Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds. As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. The conditions are satisfied by asymptotically flatmanifolds. We also prove a long time existence result for the Kähler-Ricci flow on complete nonnegatively curved Kähler manifolds.