Gradient Einstein-type warped products: rigidity, existence and nonexistence results via a nonlinear PDE

TL;DR

This paper characterizes the conditions for constructing gradient Einstein-type warped metrics, deriving a nonlinear PDE framework, and establishing nonexistence, rigidity, and construction results with explicit examples on Riemannian manifolds.

Contribution

It introduces necessary and sufficient conditions for gradient Einstein-type warped metrics, linking them to a nonlinear PDE and providing new nonexistence, rigidity, and construction results.

Findings

Derived a general Lichnerowicz equation for these metrics

Proved gradient estimates for solutions of a nonlinear elliptic PDE

Established nonexistence and rigidity results for certain warped metrics

Abstract

We establish the necessary and sufficient conditions for constructing gradient Einstein-type warped metrics. One of these conditions leads us to a general Lichnerowicz equation with analytic and geometric coefficients for this class of metrics on the space of warping functions. In this way, we prove gradient estimates for positive solutions of a nonlinear elliptic differential equation on a complete Riemannian manifold with associated Bakry-\'Emery Ricci tensor bounded from below. As an application, we provide nonexistence and rigidity results for a large class of gradient Einstein-type warped metrics. Furthermore, we show how to construct gradient Einstein-type warped metrics, and then we give explicit examples which are not only meaningful in their own right, but also help to justify our results.