Heat equation maximum principle
= Heat equation maximum principle
On a <closed manifold> with a <Riemannian metric>, the minimum of a smooth heat solution cannot fall below its initial minimum. At a spatial minimum, $\Delta u\le0$ for the <positive Laplace-Beltrami operator>. Adding a small increasing function of time and considering the first crossing gives the assertion. Applying the result to nonnegative initial functions shows that the <Riemannian heat kernel> is nonnegative.