Wave Approximation of Backward Heat Equation with Ricci Flow

Abstract

In this paper, we consider solutions of the backward heat equation with Ricci flow on manifolds as a type of infinite dimensional limit of solutions of a wave equation on a larger manifold with an analysis of wavefront set. Specifically, the projection of the solution of the wave equation $ \left(\frac{2t}{N} \cdot \frac{\partial^{2}}{\partial t^{2}} + \frac{tR(t, x)}{N} \frac{\partial}{\partial t} - \Delta_{\tilde{g}^{(N)}(t)} \right) u = R(t, x) $ outside its wavefront set onto $ \mathbb{R}_{t} \times M_{x, g(t)} $ solves the backward heat equation $ \partial_{t} u + \Delta_{x, g(t)} u = -R(t, x) $ within some appropriate time interval. We discuss this approximation starting from Euclidean case, and then extend to the open Riemannian manifold situation. This idea partially comes from Perelman's original papers in proving Poincar\'e conjecture as well as Terence Tao's Notes in UCLA.