A matrix Harnack estimate for the heat equation (Q1261727)

The work deals with positive solutions of the heat equation on compact Riemannian manifolds and generalizes a result of Li and Yau. The main result is that if the manifold \(M\) is Ricci parallel and has weakly positive sectional curvatures and if \(f\) is a positive solution to the heat equation on \(M\) for \(t>0\), then \[ D_ iD_ jf+{1\over 2t} fg_{ij}+D_ if \cdot V_ j+D_ jf \cdot V_ i+fV_ iV_ j\geq 0 \] for every vector field \(V\) on \(M\).