arXiv subjects
Publications and source records attributed to Giovanni Catino.
We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.
In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with nonnegative sectional curvature and integrable scalar curvature are quotients of the Gaussian soliton, while in the steady case we prove rigidity results under sharp integral scalar curvature decay. As a corollary, we obtain that the only three dimensional steady solitons with less than quadratic volume growth are quotients of $\mathbb{R}\timesΣ^{2}$, where $Σ^{2}$ is Hamilton's cigar.
In this paper we prove that any asymptotically cylindrical gradient shrinking Ricci soliton is isometric to a cylinder.
In this note we characterize compact hypersurfaces of dimension $n\geq 2$ with constant mean curvature $H$ immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when $n\geq 3$ and $H\neq 0$, they are locally contained in a rotational hypersurface. In dimension two, the integral pinching condition reduces to a topological assumption and we recover the classical Hopf-Chern result.
We discuss a gap in Besse's book, recently pointed out by Merton, which concerns the classification of Riemannian manifolds admitting a Codazzi tensors with exactly two distinct eigenvalues. For such manifolds, we prove a structure theorem, without adding extra hypotheses and then we conclude with some application of this theory to the classification of three-dimensional gradient Ricci solitons.
We classify compact conformally flat $n$-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either $\mathbb{S}^{n}$ with the round metric, $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ with the product metric or $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ with a rotationally symmetric Derdziński metric.
In this short note we prove that, in dimension three, flat metrics are the only complete metrics with non-negative scalar curvature which are critical for the $σ_{2}$-curvature functional.
The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three dimensional generic shrinking Ricci soliton is given by quotients of either $\mathds{S}^3$, $\erre\times\mathds{S}^2$ or $\erre^3$, under some very weak conditions on the vector field $X$ generating the soliton structure. In doing so we introduce analytical tools that could be useful in other settings; for instance we prove that the Omori-Yau maximum principle holds for the $X$-Laplacian on every generic Ricci soliton, without any assumption on $X$.
In this paper we introduce, in the Riemannian setting, the notion of conformal Ricci soliton, which includes as particular cases Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons. We provide here some necessary integrability conditions for the existence of these structures that also recover, in the corresponding contexts, those already known in the literature for conformally Einstein manifolds and for gradient Ricci solitons. A crucial tool in our analysis is the construction of some appropriate and highly nontrivial $(0,3)$-tensors related to the geometric structures, that in the special case of gradient Ricci solitons become the celebrated tensor $D$ recently introduced by Cao and Chen. A significant part of our investigation, which has independent interest, is the derivation of a number of commutation rules for covariant derivatives (of functions and tensors) and of transformation laws of some geometric objects under a conformal change of the underlying metric.
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.
In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold $\varrg$, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen.
In this paper we consider a perturbation of the Ricci solitons equation proposed in \cite{jpb1} and studied in \cite{CaMa} and we classify noncompact gradient shrinkers with bounded nonnegative sectional curvature.
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.
We compute the evolution equation of the Weyl tensor under the Ricci flow of a Riemannian manifold and we discuss some consequences for the classification of locally conformally flat Ricci solitons.
In this paper we investigate complete critical metrics of the $L^{2}$-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.
In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product $\mathbb{R}\times N^{n-1}$, or globally conformally equivalent to the Euclidean space $\mathbb{R}^{n}$ or to the round sphere $\mathbb{S}^{n}$. In particular, we show that any complete, noncompact, gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric, whenever the potential function is nonconstant.
In this paper we prove that any $n$-dimensional ($n\ge 4$) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in [6] and [9].
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.