Optimal bounds for ancient caloric functions

Abstract

For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions on any space where Yau's 1974 conjecture about polynomial growth harmonic functions holds.

arXiv commentA stronger sharp dimension bound is added which is an equality on Euclidean space. To appear in Duke Math. Journal