Gradient Einstein solitons

TL;DR

This paper studies a perturbation of Ricci solitons called gradient Einstein solitons, proving their rigidity and classifying the unique complete three-dimensional example with positive curvature, which is rotationally symmetric and asymptotically cylindrical.

Contribution

It introduces and analyzes a perturbed Ricci soliton equation, establishing rigidity results and classifying the unique three-dimensional solution with positive curvature.

Findings

Only one such complete 3D manifold exists with positive curvature.

The manifold is rotationally symmetric and asymptotically cylindrical.

The potential function's rectifiability is key to the proof.

Abstract

In this paper we consider a perturbation of the Ricci solitons equation proposed by J. P. Bourguignon in \cite{jpb1}. We show that these structures are more rigid then standard Ricci solitons. In particular, we prove that there is only one complete three--dimensional, positively curved, Riemannian manifold satisfying $$ Ric -\frac{1}{2} R \, g \, + \, \nabla^2 f \, = \,0\,, $$ for some smooth function $f$. This solution is rotationally symmetric and asymptotically cylindrical and it represents the analogue of the Hamilton's cigar in dimension three. The key ingredient in the proof is the rectifiability of the potential function $f$. It turns out that this property holds also in the Lorentzian setting and for a more general class of structures which includes some gravitational theories.