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Stefano Borghini

Publications and source records attributed to Stefano Borghini.

16 recordsLinked to original sources
arXiv2026

Estimates for the p-Green function near the pole

We study the asymptotic expansion of p-Green functions and their derivatives near the pole. In the Euclidean setting, we strengthen a theorem from [arXiv:2203.01206] by establishing improved integrability properties and estimates for the first and second derivatives. In the more general Riemannian setting, we derive a refined asymptotic expansion of the p-Green function near its pole, thereby improving the corresponding result proved in [arXiv:2512.14591].

math.AP↗
arXiv2026

Mass-type invariants in the presence of a cosmological constant

In this paper, we introduce a new family of mass-type invariants for time-symmetric initial data in space-times satisfying the Dominant Energy Condition. For positive cosmological constant, these invariants, unlike the total Hawking mass, turn out to be genuinely effective in providing new characterizations of the de Sitter solution. From a theoretical standpoint, this opens a new perspective on how one might refine the rigidity statement originally proposed by Min-Oo in his well known conjecture, later refuted by the counterexamples of Brendle, Marques, and Neves. Via a formal limiting procedure, we also define another invariant, the 1-harmonic Mass, for which we independently prove a positive mass theorem and a Penrose-type inequality, thereby extending tools for probing space-time geometries in the presence of a positive cosmological constant.

math.DG↗
arXiv2024

Black Hole and Equipotential Photon Surface Uniqueness in 4-dimensional Asymptotically Flat Electrostatic Electro-Vacuum Spacetimes

We study 4-dimensional asymptotically flat electrostatic electro-vacuum spacetimes with a connected black hole, photon sphere, or equipotential photon surface inner boundary. Our analysis, inspired by the potential theory approach by Agostiniani-Mazzieri, allows to give self-contained proofs of known uniqueness theorems of the sub-extremal, extremal and super-extremal Reissner-Nordström spacetimes. We also obtain new results for connected photon spheres and for connected photon surfaces in the extremal case. Finally, we provide, up to a restriction on the range of their radii, the uniqueness result for connected (both non-degenerate and degenerate) photon surfaces in the super-extremal case, not yet treated in the literature.

gr-qc↗
arXiv2023

Comparison geometry for substatic manifolds and a weighted Isoperimetric Inequality

Substatic Riemannian manifolds with minimal boundary arise naturally in General Relativity as spatial slices of static spacetimes satisfying the Null Energy Condition. Moreover, they constitute a vast generalization of nonnegative Ricci curvature. In this paper we will prove various geometric results in this class, culminating in a sharp, weighted Isoperimetric inequality that quantifies the area minimizing property of the boundary. Its formulation and proof will build on a comparison theory partially stemming from a newly discovered conformal connection with $\mathrm{CD}(0, 1)$ metrics.

math.DG↗
arXiv2023

The equality case in the substatic Heintze-Karcher inequality

We provide a rigidity statement for the equality case for the Heintze-Karcher inequality in substatic manifolds. We apply such result in the warped product setting to fully remove assumption (H4) in the celebrated Brendle's characterization of constant mean curvature hypersurfaces in warped products.

math.DG↗
arXiv2023

Counterexamples to a divergence lower bound for the covariant derivative of skew-symmetric 2-tensor fields

In arXiv:2103.15482 an estimate for suitable skew-symmetric 2-tensors was claimed. Soon after, this estimate has been exploited to claim powerful classification results: most notably, it has been employed to propose a proof of a Black Hole Uniqueness Theorem for vacuum static spacetimes with positive scalar curvature and in connection with the Besse Conjecture. In the present note we point out an issue in the argument proposed in arXiv:2103.15482 and we provide a counterexample to the estimate.

math.DG↗
arXiv2021

On the Serrin problem for ring-shaped domains

In this paper, we deal with the long standing open problem of characterising rotationally symmetric solutions to $Δu = -2$, when Dirichlet boundary conditions are imposed on a ring-shaped planar domain. From a physical perspective, the solution represents the velocity of a homogeneous incompressible fluid, flowing in steady parallel streamlines through a hollow cylindrical pipe and obeying a no-slip condition. In contrast with Serrin's classical result, we show that the simplest possible set of overdetermining conditions, namely the prescription of locally constant Neumann boundary data, is not sufficient to obtain a complete characterisation of the solutions. A further requirement on the number of maximum points arises in our analysis as a necessary and sufficient condition for the rotational symmetry. In fluid-dynamical terms, our results imply that if the wall shear stress is constant on some connected component of the pipe's wall, then the velocity of the fluid must attain its maximal value only at finitely many streamlines, unless the hollow pipe itself consists of a couple of concentric cylindrical round tubes. A major difficulty in the analysis of this problem comes from the lack of monotonicity of the model solutions, which makes the moving plane method ineffective. To remedy this issue, we introduce some new arguments in the spirit of comparison geometry, that we believe of independent interest.

math.AP↗
arXiv2020

Static Black Hole Uniqueness for nonpositive masses

In arXiv:1310.3002, Lee and Neves proved a Penrose inequality for spacetimes with negative cosmological constant and nonpositive mass aspect. As an application, they were able to obtain a static uniqueness theorem for Kottler spacetimes with nonpositive mass. In this paper, we propose an alternative more elementary proof of this static uniqueness result and we discuss some possible generalizations, most notably to degenerate horizons.

math.DG↗
arXiv2019

On the uniqueness of Schwarzschild-de Sitter spacetime

We establish a new uniqueness theorem for the three dimensional Schwarzschild-de Sitter metrics. For this some new or improved tools are developed. These include a reverse Lojasiewicz inequality, which holds in a neighborhood of the extremal points of any smooth function. We further prove smoothness of the set of maxima of the lapse, whenever this set contains a topological hypersurface. This leads to a new strategy for the classification of well behaved static solutions of Einstein equations with a positive cosmological constant, based on the geometry of the maximum-set of the lapse.

math.DG↗
arXiv2019

On the mass of static metrics with positive cosmological constant -- II

This is the second of two works, in which we discuss the definition of an appropriate notion of mass for static metrics, in the case where the cosmological constant is positive and the model solutions are compact. In the first part, we have established a positive mass statement, characterising the de Sitter solution as the only static vacuum metric with zero mass. In this second part, we prove optimal area bounds for horizons of black hole type and of cosmological type, corresponding to Riemannian Penrose inequalities and to cosmological area bounds à la Boucher-Gibbons-Horowitz, respectively. Building on the related rigidity statements, we also deduce a uniqueness result for the Schwarzschild--de Sitter spacetime.

math.AP↗
arXiv2017

On the mass of static metrics with positive cosmological constant - I

In this paper we propose and discuss a notion of mass for compact static metrics with positive cosmological constant. As a consequence, we characterise the de Sitter solution as the only static vacuum metric with zero mass. Finally, we show how to adapt our analysis to the case of negative cosmological constant, leading to a uniqueness theorem for the Anti de Sitter spacetime.

math.AP↗
arXiv2017

Some Sphere Theorems in Linear Potential Theory

In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular bounded domain $Ω\subset \mathbb{R}^n$, $n\geq 3$, we prove that if the mean curvature $H$ of the boundary obeys the condition $$ - \bigg[ \frac{1}{\text{Cap}(Ω)} \bigg]^{\frac{1}{n-2}} \leq \frac{H}{n-1} \leq \bigg[ \frac{1}{\text{Cap}(Ω)} \bigg]^{\frac{1}{n-2}} , $$ then $Ω$ is a round ball.

math.DG↗
arXiv2016

Monotonicity formulas for static metrics with non-zero cosmological constant

In this paper we study non-singular vacuum static space-times with non-zero cosmological constant. We introduce new integral quantities, and under suitable assumptions we prove their monotonicity along the level set flow of the static potential. We then show how to use these properties to derive a number of sharp geometric and analytic inequalities, whose equality case can be used to characterize the rotational symmetry of the underlying static solutions. As a consequence, we are able to prove some new uniqueness statements for the de Sitter and the anti-de Sitter metrics. In particular, we show that the de Sitter solution has the least possible surface gravity among three-dimensional static metrics with connected boundary and positive cosmological constant.

math.DG↗
arXiv2014

A gluing formula for Reidemeister-Turaev torsion

We extend Turaev's theory of Euler structures and torsion invariants on 3-manifolds to the case of vector fields having generic behavior on the boundary. This allows to easily define gluings of Euler structures and to develop a completely general gluing formula for Reidemeister torsion of 3-manifolds. Lastly, we describe a combinatorial presentation of Euler structures via stream-spines, as a tool to effectively compute torsion.

math.GT↗