Matrix Harnack estimate for the heat equation

Abstract

In the important paper[LY] by Peter Li and S.-T. Yau, they show how the classical Harnack principle for the heat equation on a manifold can be derived from a differential inequality. In particular, they show that for any positive solution / > 0 of the heat equation dt J on a compact Riemannian manifold of dimension m solving the equation for t > 0, if the manifold has weakly positive Ricci curvature ijy > 0 then for any vector field V on M ?t + Lf + 2Df(V) + f\V\*>0 ] a similar result holds with an error term if the Ricci curvature is bounded below. The quadratic version in V given here is equivalent to the more complicated formula in their paper by choosing the optimal V. We shall show in this paper that the Harnack estimate of Li and Yau is the trace of a full matrix inequality. Main Theorem. If M is a compact Riemannian manifold and f > 0 is a positive solution to the heat equation on M dt J for t > 0, then for any vector field V* on M we have DiDjf + ^ fga + A/ • ^ + 2V • K + fV^ > 0 Reseaxch partially supported by NSF contract DMS 90-03333.