arXiv subjects

Paolo Mastrolia

Publications and source records attributed to Paolo Mastrolia.

At least 19 recordsLinked to original sources

arXiv2026

Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$

We describe a method to prove new integral inequalities for stable minimal hypersurfaces in Euclidean space. As an application, we give a simple proof that complete, two sided, stable minimal hypersurfaces in $\mathbb{R}^4$ are hyperplanes. A core part of the argument hinges on the fact that stable minimal hypersurfaces in non-negatively curved spaces are examples of manifolds with a spectral Ricci curvature lower bound; in particular, we prove a sharp pointwise gradient estimate for the Green kernel on non-parabolic manifolds with spectral Ricci lower bounds, extending previous work by Colding.

math.DG↗

arXiv2026

Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces

In this paper we prove general criticality criteria for operators $Δ+ V$ on manifolds with more than one end, where $V$ bounds the Ricci curvature, and a related spectral splitting theorem extending Cheeger-Gromoll's one. Our results give new insight on Li-Wang's theory of manifolds with a weighted Poincaré inequality. We apply them to study stable and $δ$-stable minimal hypersurfaces in manifolds with non-negative bi-Ricci or sectional curvature, in ambient dimension up to $5$ and $6$, respectively. In the special case where the ambient space is $\mathbb{R}^4$, we prove that a $1/3$-stable minimal hypersurface must either have one end or be a catenoid, and that proper, $δ$-stable minimal hypersurfaces with $δ> 1/3$ must be hyperplanes.

math.DG↗

arXiv2025

On the Geometry of $φ$-Static Perfect Fluid Space-Times

In this paper we study the geometry of $φ$-static perfect fluid space-times ($φ$-SPFST, for short). In the context of Einstein's General Relativity, they arise from a space-time whose matter content is described by a perfect fluid in addition to a nonlinear field expressed by a smooth map $φ$ with values in a Riemannian manifold. Considering the Lorentzian manifold $\hat{M}$ in the form of a static warped product, we derive the fundamental equations via reduction of Einstein's Field Equations to the factors of the product. To set the stage for our main results, we discuss the validity of the classical Energy Conditions in the present setting and we introduce the formalism of $φ$-curvatures, which is a fundamental tool to merge the geometry of the manifold with that of the smooth map $φ$. We then present several mathematical settings in which similar structures arise. After computing two integrability conditions, we apply them to prove a number of rigidity results, both for manifolds with or without boundary. In each of the aforementioned results, the main assumption is given by the vanishing of some $φ$-curvature tensors and the conclusion is a local splitting of the metric into a warped product. Inspired by the classical Cosmic No Hair Conjecture of Boucher, Gibbons and Horowitz, we find sharp sufficient conditions on a compact $φ$-SPFST with boundary to be isometric to the standard hemisphere. We then describe the geometry of relatively compact domains in $M$ subject to an upper bound on the mean curvature of their boundaries. Finally, we study non-existence results for $φ$-SPFSTs, both via the existence of zeroes of the solutions of an appropriate ODE and with the aid of a suitable integral formula generalising in a precise sense the well-known Kazdan-Warner obstruction.

math.DG↗

arXiv2025

Some geometric properties of generalized $φ$-vacuum static spaces

In this paper we study the geometry of generalized $φ$-vacuum static spaces, proving estimates for the $φ$-scalar curvature and for the first eigenvalue of the Jacobi operator, and also rigidity under various geometric assumptions; in particular, we prove a result related to the famous Cosmic no-hair conjecture of Boucher, Gibbons and Horowitz.

math.DG↗

arXiv2025

Uniqueness of critical metrics for a quadratic curvature functional

In this paper we prove a new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functional $\mathfrak{S}^2 = \int R_g^{2} dV_g$: we show that critical metrics $(M^n, g)$ with finite energy are always scalar flat, i.e. global minima, provided $n\geq 10$.

math.DG↗

arXiv2024

On Riemannian four-manifolds and their twistor spaces: a moving frame approach

In this paper we study the twistor space $Z$ of an oriented Riemannian four-manifold $M$ using the moving frame approach, focusing, in particular, on the Einstein, non-self-dual setting. We prove that any general first-order linear condition on the almost complex structures of $Z$ forces the underlying manifold $M$ to be self-dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first-order quadratic conditions, showing that the Atiyah-Hitchin-Singer almost Hermitian twistor space of an Einstein four-manifold bears a resemblance, in a suitable sense, to a nearly Kähler manifold.

math.DG↗

arXiv2024

Some canonical metrics {\em via} Aubin's local deformations

In this paper, using special metric deformations introduced by Aubin, we construct Riemannian metrics satisfying non-vanishing conditions concerning the Weyl tensor, on every compact manifold. In particular, in dimension four, we show that there are no topological obstructions for the existence of metrics with non-vanishing Bach tensor.

math.DG↗

arXiv2024

Rigidity results for Riemannian twistor spaces under vanishing curvature conditions

In this paper we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces.In particular, using the moving frame method, we prove that $\mathbb{CP}^3$ is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold.

math.DG↗

arXiv2024

Rigidity of Einstein manifolds with positive Yamabe invariant

We provide optimal pinching results on closed Einstein manifolds with positive Yamabe invariant in any dimension, extending the optimal bound for the scalar curvature due to Gursky and LeBrun in dimension four. We also improve the known bounds of the Yamabe invariant \emph{via} the $L^{\frac{n}{2}}$-norm of the Weyl tensor for low-dimensional Einstein manifolds. Finally, we discuss some advances on an algebraic inequality involving the Weyl tensor for dimensions $5$ and $6$.

math.DG↗

arXiv2023

Two rigidity results for stable minimal hypersurfaces

The aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in $\mathbb{R}^4$, while they do not exist in positively curved closed Riemannian $(n+1)$-manifold when $n\leq 5$; in particular, there are no stable minimal hypersurfaces in $\mathbb{S}^{n+1}$ when $n\leq 5$. The first result was recently proved also by Chodosh and Li, and the second is a consequence of a more general result concerning minimal surfaces with finite index. Both theorems rely on a conformal method, inspired by a classical work of Fischer-Colbrie.

math.DG↗

arXiv2021

Rigidity of Critical Metrics for Quadratic Curvature Functionals

In this paper we prove new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functionals $\mathfrak{F}^{2}_t = \int |\operatorname{Ric}_g|^{2} dV_g + t \int R^{2}_g dV_g$, $t\in\mathbb{R}$, and $\mathfrak{S}^2 = \int R_g^{2} dV_g$. We show that (i) flat surfaces are the only critical points of $\mathfrak{S}^2$, (ii) flat three-dimensional manifolds are the only critical points of $\mathfrak{F}^{2}_t$ for every $t>-\frac{1}{3}$, (iii) three-dimensional scalar flat manifolds are the only critical points of $\mathfrak{S}^2$ with finite energy and (iv) $n$-dimensional, $n>4$, scalar flat manifolds are the only critical points of $\mathfrak{S}^2$ with finite energy and scalar curvature bounded below. In case (i), our proof relies on rigidity results for conformal vector fields and an ODE argument; in case (ii) we draw upon some ideas of M. T. Anderson concerning regularity, convergence and rigidity of critical metrics; in cases (iii) and (iv) the proofs are self-contained and depend on new pointwise and integral estimates.

math.DG↗

arXiv2019

A conformal Yamabe problem with potential on the euclidean space

We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.

math.DG↗

arXiv2018

Four dimensional closed manifolds admit a weak harmonic Weyl metric

On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by a reference metric, constructed by Aubin, with nowhere vanishing Weyl tensor. Moreover, we show that anti-self-dual metrics with positive Yamabe invariant can be characterized by pinching conditions involving suitable quadratic Riemannian functionals.

math.DG↗

arXiv2018

Weyl scalars on compact Ricci solitons

We investigate the triviality of compact Ricci solitons under general scalar conditions involving the Weyl tensor. More precisely, we show that a compact Ricci soliton is Einstein if a generic linear combination of divergences of the Weyl tensor contracted with suitable covariant derivatives of the potential function vanishes. In particular we recover and improve all known related results. This paper can be thought as a first, preliminary step in a general program which aims at showing that Ricci solitons can be classified finding a "generic" $[k, s]$-vanishing condition on the Weyl tensor, for every $k, s\in\mathbb{N}$, where $k$ is the order of the covariant derivatives of Weyl and $s$ is the type of the (covariant) tensor involved.

math.DG↗

arXiv2017

A potential generalization of some canonical Riemannian metrics

The aim of this paper is to study new classes of Riemannian manifolds endowed with a smooth potential function, including in a general framework classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and, above all, gradient Ricci solitons. For the most rigid cases we give a complete classification, while for the others we provide rigidity and obstruction results, characterizations and nontrivial examples. In the final part of the paper we also describe the "nongradient" version of this construction.

math.DG↗

arXiv2016

Bochner type formulas for the Weyl tensor on four dimensional Einstein manifolds

The very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner type formulas for the Weyl tensor on a four dimensional Einstein manifold. In particular, we prove a second Bochner type formula which, formally, extends to the covariant derivative level the classical one for the Weyl tensor obtained by Derdzinski in 1983. As a consequence, we deduce some integral identities involving the Weyl tensor and its derivatives on a compact four dimensional Einstein manifold.

math.DG↗

arXiv2016

Gradient Ricci solitons with vanishing conditions on Weyl

We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any $n$-dimensional ($n\geq 4$) gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite quotient of $N^{n-k}\times \mathbb{R}^k$, $(k > 0)$, the product of a Einstein manifold $N^{n-k}$ with the Gaussian shrinking soliton $\mathbb{R}^k$. The technique applies also to the steady and expanding cases in all dimensions. In particular, we prove that a three dimensional gradient steady soliton with third order divergence-free Cotton tensor, i.e. with vanishing double divergence of the Bach tensor, is either flat or isometric to the Bryant soliton.

math.DG↗

arXiv2015

Elliptic and parabolic equations with Dirichlet conditions at infinity on Riemannian manifolds

We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is studied. We consider also elliptic equations on $M$ with similar conditions at infinity.

math.AP↗