Liouville theorems for ancient caloric functions via optimal growth conditions

TL;DR

This paper establishes Liouville theorems for ancient solutions to the heat equation on non-compact manifolds, identifying optimal growth conditions that guarantee solutions are trivial.

Contribution

It introduces new Liouville theorems for ancient caloric functions under optimal growth conditions on manifolds with Ricci curvature bounds.

Findings

Growth conditions ensuring trivial solutions are optimal.

Liouville theorems hold under specified curvature and growth assumptions.

Examples demonstrate the sharpness of the conditions.

Abstract

We provide some Liouville theorems for ancient nonnegative solutions of the heat equation on a complete non-compact Riemannian manifold with Ricci curvature bounded from below. We determine growth conditions ensuring triviality of the latters, showing their optimality through examples.