Gradient Einstein solitons

By G. Catino, L. Mazzieri. 2012

146 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/50692c7302343864b612728305338b950fd40484

Citations in the corpus

• The Ricci-Bourguignon Flow (2015, unknown): “Some results concerning solitons of the Ricci–Bourguignon flow (called gradient ρ -Einstein solitons ) can be found in [Catino and Mazzieri 2016; Catino et al. 2015b].”