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

TL;DR

This work develops a comprehensive framework for gradient Einstein-type warped products, unifying and extending several soliton and Einstein-type structures through nonlinear PDEs. The authors derive necessary and sufficient base/fiber conditions, culminating in a Lichnerowicz-type equation for the warping function and a drift-Laplacian framework that governs existence and rigidity. A key contribution is a local gradient estimate in the Bakry–Émery setting, which yields rigidity and nonexistence results for broad classes of gradient Einstein-type warped products, including Ricci solitons and Einstein cases, as well as explicit construction methods via symmetry-based ansatz. The results illuminate when warped products must reduce to standard products and provide explicit constructions and sharp growth conditions, linking geometric analysis with PDE techniques to advance the theory of Einstein-type manifolds.

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-Émery 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.

Theorems & Definitions (8)

• proof : Proof of Proposition \ref{['thm:wpgEtM-and-lfB']}
• proof
• proof
• proof : Proof of Theorem \ref{['thm:gEtM-c']}
• proof : Proof of Theorem \ref{['Th2']}
• proof : Proof of Corollary \ref{['Cor1-triviality']}
• proof : Proof of Corollary \ref{['Cor2-nonexistence']}
• proof : Proof of Corollary \ref{['sec:discussions-3']}