Rigidity for the heat equation with density on Riemannian manifolds through a conformal change

TL;DR

This paper establishes conditions for the uniqueness of solutions to the heat equation with density on weighted Riemannian manifolds, using conformal transformations to analyze solution vanishing and demonstrating optimality through counterexamples.

Contribution

It introduces a unified method based on conformal changes to determine solution uniqueness for the heat equation with density on weighted manifolds.

Findings

Identifies sufficient conditions for solution vanishing in weighted Lebesgue spaces.

Develops a conformal transformation technique to analyze the heat equation.

Provides counterexamples confirming the optimality of assumptions.

Abstract

We investigate uniqueness of solution to the heat equation with a density $\rho$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_{\phi}$. We distinguish between the cases $p > 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $\rho$. We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $\rho$.