Gradient Estimates on Warped Product Gradient Almost Ricci Solitons

TL;DR

This paper develops new gradient estimates for warping functions on warped product gradient almost Ricci solitons by adapting Li-Yau's technique to drifting Laplacians, leading to nonexistence results under specific conditions.

Contribution

It introduces three novel gradient estimates for the warping function, tailored to the sign of the Einstein constant, and applies these to establish nonexistence theorems.

Findings

Three gradient estimates for the warping function based on Einstein constant sign

A nonexistence theorem for certain warped product gradient almost Ricci solitons

Modified Li-Yau technique for handling drifting Laplacians

Abstract

In this paper, by slightly modifying Li-Yau's technique so that we can handle drifting Laplacians, we were able to find three different gradient estimates for the warping function, one for each sign of the Einstein constant of the fiber manifold. As an application, we exhibit a nonexistence theorem for gradient almost Ricci solitons possessing certain metric properties on the base of the warped product.