For any manifold with polynomial volume growth, we show that 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.
"Optimal bounds for ancient caloric functions." Duke Math. J. 170 (18) 4171 - 4182, 1 December 2021. https://doi.org/10.1215/00127094-2021-0015