Sobolev-type inequalities and heat kernel bounds along the geometric flow

This paper is concerned with Sobolev-type inequalities and upper bound for the fundamental solution to the heat-type equation defined on compact manifold whose metric evolves by the generalized geometric flow. It turns out that the pointwise estimates obtained in this paper depend on the constants in the uniform Sobolev inequalities for the flow or the best constants in the euclidean Sobolev embedding. We give various illustrations to show that our results are valid in many contexts of geometric flow, where we may not need explicit curvature constraint. Moreover, our approach here also demonstrates equivalence of Sobolev inequalities, log-Sobolev inequalities, ultracontractive estimates and heat kernel upper bounds.