Calculus of Variations and Geometric Measure Theory
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G. Catino - L. Mazzieri

Gradient Einstein solitons

created by catino on 01 Feb 2012
modified on 10 Apr 2016


Published Paper

Inserted: 1 feb 2012
Last Updated: 10 apr 2016

Journal: Nonlinear Anal.
Volume: 132
Pages: 66-94
Year: 2016


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 $$ Ric -\frac{1}{2} R \, g \, + \, \nabla2 f \, = \,0\,, $$ 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.


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