Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow

TL;DR

This paper derives gradient estimates for a parabolic PDE linked to the Ricci-Bourguignon flow on warped product manifolds, extending classical heat equation results and providing explicit solutions.

Contribution

It introduces new gradient estimates for PDEs under Ricci-Bourguignon flow and constructs explicit warped solutions demonstrating the theory.

Findings

Established gradient estimates for the PDE under Ricci-Bourguignon flow.

Recovered classical gradient estimates for the heat equation under Ricci flow.

Constructed explicit examples of warped solutions illustrating the results.

Abstract

We study the Ricci-Bourguignon flow on warped product manifolds with noncompact base. This setting leads naturally to a parabolic partial differential equation on the space of smooth warping functions, arising from the necessary and sufficient conditions for a warped metric to evolve under the flow. One of our main results establishes a gradient estimate for this equation, providing the analytic input for the geometric applications developed herein and, in particular, recovering classical gradient estimates for the heat equation under the Ricci flow. Furthermore, we show how to construct explicit warped solutions to the Ricci-Bourguignon flow and present examples that are not only of independent interest but also illustrate and support our results