A Pohozaev Identity on Warped Product Solitons

TL;DR

This paper establishes a Pohozaev identity for warped product shrinking gradient Ricci solitons, showing under certain conditions that such metrics must be simple cross products, paralleling results in reaction-diffusion equations.

Contribution

It introduces a Pohozaev-type identity for warped product Ricci solitons and proves a rigidity result for two-dimensional bases with bounded curvature.

Findings

Warped product Ricci solitons with bounded curvature and 2D base are cross products.

The Pohozaev identity leads to a rigidity theorem for these solitons.

Results parallel blow-up analysis in reaction-diffusion equations.

Abstract

Warped product metrics are a class of Riemannian metrics on cross products $B \times F$ which have been well studied and provide a rich set of examples. In this paper we consider shrinking gradient Ricci solitons which are warped product metrics. We prove that if the curvature of the metric is bounded and the base $B$ has two dimensions, the metric must in fact be a cross product. The theorem is a consequence of a Pohozaev-type identity. This closely parallels results in the study of blow-ups of solutions to the reaction- diffusion equation $\partial_t v = \Delta v + e^v$ in $\mathbb{R}^2$ .