# Optimal bounds for ancient caloric functions

## TL;DR

This paper establishes bounds on the dimension of ancient caloric functions with polynomial growth on manifolds, linking it to harmonic functions and confirming sharp bounds under Yau's conjecture.

## Contribution

It provides the first bounds for ancient caloric functions on manifolds with polynomial volume growth, connecting them to harmonic functions and Yau's conjecture.

## Key findings

- Bound on the dimension of ancient caloric functions proportional to polynomial growth degree
- Sharp bounds achieved under Yau's 1974 conjecture
- Extension of harmonic function results to 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.