Dimensional bounds for ancient caloric functions on graphs

TL;DR

This paper extends a theorem from manifolds to graphs, establishing bounds on ancient caloric functions of polynomial growth based on harmonic functions and volume growth.

Contribution

It generalizes Colding and Minicozzi's theorem from manifolds to graphs, providing bounds on ancient solutions of heat equations on graphs.

Findings

Bound on the dimension of ancient caloric functions on graphs

Extension of manifold results to graph settings

Relation between harmonic functions and caloric solutions

Abstract

We study ancient solutions of polynomial growth to heat equations on graphs, and extend Colding and Minicozzi's theorem [CM19] on manifolds to graphs: For a graph of polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the product of the growth degree and the dimension of harmonic functions with the same growth.