On noncompact warped product Ricci solitons

TL;DR

This paper investigates complete noncompact warped product gradient Ricci solitons, establishing nonexistence results, estimates, and rigidity properties, with special focus on steady, expanding, and shrinking cases.

Contribution

It provides new nonexistence theorems and estimates for warped product Ricci solitons, extending known results and introducing novel results for shrinking solitons.

Findings

Nonexistence results for steady and expanding solitons

Estimates for the warping function and its gradient

A new nonexistence theorem for shrinking solitons

Abstract

The goal of this article is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain pde estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian.