On the nonexistence of quasi-Einstein metrics

TL;DR

This paper proves the nonexistence of nonconstant solutions to certain quasi-Einstein equations on complete manifolds, leading to new rigidity results for Ricci-flat warped products and steady gradient Ricci solitons.

Contribution

It establishes the nonexistence of nonconstant solutions to the PDE associated with quasi-Einstein metrics and derives implications for Ricci-flat warped products and Ricci solitons.

Findings

No nonconstant solutions to the PDE for $ abla^2 f$ equations.

Nonexistence of nontrivial Ricci flat warped products with nonpositive Einstein fiber.

Constant $R + | abla f|^2$ for nontrivial steady gradient Ricci solitons.

Abstract

We study complete Riemannian manifolds satisfying the equation $Ric+\nabla^2 f-\frac{1}{m}df\otimes df=0$ by studying the associated PDE $\Delta_f f + m\mu e^{2f/m}=0$ for $\mu\leq 0$. By developing a gradient estimate for $f$, we show there are no nonconstant solutions. We then apply this to show that there are no nontrivial Ricci flat warped products with fibers which have nonpositive Einstein constant. We also show that for nontrivial steady gradient Ricci solitons, the quantity $R+|\nabla f|^2$ is a positive constant.