Space of ancient caloric functions on some manifolds beyond volume doubling

TL;DR

This paper establishes a polynomial structure for ancient caloric functions on certain manifolds without volume doubling, extending classical results on harmonic functions and removing local geometric restrictions.

Contribution

It introduces a new condition that surpasses volume doubling, leading to a polynomial structure theorem for ancient caloric functions on manifolds.

Findings

Finiteness results for harmonic functions are sharp except for multi-end cases

Removed local topological or geometric conditions on manifolds

Extended classical polynomial growth results to broader manifold classes

Abstract

Under a condition that breaks the volume doubling barrier, we obtain a time polynomial structure result on the space of ancient caloric functions with polynomial growth on manifolds. As a byproduct, it is shown that the finiteness result for the space of harmonic functions with polynomial growth on manifolds in \cite{CM97} and \cite{Li97} are essentially sharp, except for the multi-end cases, addressing an issue raised in \cite{CM98} and removing all {\it local} topological or geometric conditions on the manifold with respect to a reference point.