131 research outputs found
Some rigidity results on critical metrics for quadratic functionals
In this paper we prove rigidity results on critical metrics for quadratic
curvature functionals, involving the Ricci and the scalar curvature, on the
space of Riemannian metrics with unit volume. It is well-known that Einstein
metrics are always critical points. The purpose of this article is to show
that, under some curvature conditions, a partial converse is true. In
particular, for a class of quadratic curvature functionals, we prove that every
critical metric with non-negative sectional curvature must be Einstein
Critical metrics of the -norm of the scalar curvature
In this paper we investigate complete critical metrics of the -norm of
the scalar curvature. We prove that any complete critical metric with positive
scalar curvature has constant scalar curvature and we characterize critical
metrics with nonnegative scalar curvature in dimension three and four
Integral pinched shrinking Ricci solitons
We prove that a -dimensional, , compact gradient
shrinking Ricci soliton satisfying a -pinching condition is isometric
to a quotient of the round . The proof relies mainly on sharp
algebraic curvature estimates, the Yamabe-Sobolev inequality and an improved
rigidity result for integral pinched Einstein metrics
A note on four dimensional (anti-)self-dual quasi-Einstein manifolds
In this short note we prove that any complete four dimensional anti-self-dual
(or self-dual) quasi-Einstein manifolds is either Einstein or locally
conformally flat. This generalizes a recent result of X. Chen and Y. Wang
A Weyl Entropy of Pure Spacetime Regions
We focus on the Penrose's Weyl Curvature Hypothesis in a general framework
encompassing many specific models discussed in literature. We introduce a
candidate density for the Weyl entropy in pure spacetime perfect fluid regions
and show that it is monotonically increasing in time under very general
assumptions. Then we consider the behavior of the Weyl entropy of compact
regions, which is shown to be monotone in time as well under suitable
hypotheses, and also maximal in correspondence with vacuum static metrics. The
minimal entropy case is discussed too
- …