*Published Paper*

**Inserted:** 3 jul 2017

**Last Updated:** 17 oct 2023

**Journal:** Nonlinear Anal.

**Volume:** 132

**Pages:** 66-94

**Year:** 2016

**Doi:** 10.1016/j.na.2015.10.021

**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.

**Tags:**
FIRB2012-DGGFT