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Benedito Leandro

Publications and source records attributed to Benedito Leandro.

At least 19 recordsLinked to original sources
arXiv2026

Semilinear overdetermined problems, a divergence formula, and geometric inequalities

Considering an $n$-dimensional compact Riemannian manifold with a boundary that satisfies a Serrin-type problem, we prove sharp upper and lower bounds for the area of such a boundary. Then, we present the main result of this paper: a divergence formula for a special vector field on a given Riemannian manifold. This divergence formula is closely related to sub-static manifolds and related metrics, e.g., $V$-static, static, and electrostatic manifolds. We show Minkowski-type inequalities for $V$-sub-static manifolds under different Neumann boundary conditions. Moreover, we prove an area-charge inequality for compact (and noncompact) electrostatic manifolds. The main application of the divergence formula presented in this work generalizes the classical result of Boucher--Gibbons--Horowitz: we prove that an asymptotically hyperbolic space of a static spacetime satisfying the null convergence condition must be the hyperbolic space.

math.DG↗
arXiv2026

Solution for the Einstein-Maxwell equations invariant under an $(n - 1)$-dimensional group of dilations

We consider an electrostatic system whose spatial factor is conformal to an $n$-dimensional Euclidean space. We provide a complete characterization of the most general ansatz, thereby reducing the associated electrostatic system of partial differential equations to an ordinary differential equation system. We prove that there are only two possibilities: either the cosmological constant is nonzero, in which case the solutions are necessarily invariant under rotations or translations, or the cosmological constant vanishes, and the solutions belong to the Majumdar-Papapetrou class with a degree of freedom associated with an invariant $(n-1)$-dimensional subgroup. As a result, we introduce a new solution to the electrovacuum system in the Majumdar-Papapetrou class that is invariant under an $(n-1)$-dimensional group of dilations.

math.DG↗
arXiv2025

Geometric inequalities for electrostatic systems with boundary

In this article, we investigate electrostatic systems with a nonzero cosmological constant on compact manifolds with boundary. We establish new geometric properties for electrostatic manifolds in higher dimensions, extending previous results in the literature. Moreover, we prove sharp boundary estimates and isoperimetric-type inequalities for electrostatic manifolds, as well as volume and boundary inequalities involving the Brown-York and Hawking masses.

math.DG↗
arXiv2025

Sharp lower bound for the charged Hawking mass in the electrostatic space

We prove sharp lower bounds for the charged Hawking mass of stable surfaces in electrostatic space-times in various contexts. An upper bound for the genus of stable surfaces in the electrostatic system is provided. We also study the positivity for the charged Hawking mass of a minimal surface with index one in the electrostatic space-times. A criterion for a CMC surface in the Reissner-Nordstrom deSitter space to be stable is presented.

math.DG↗
arXiv2024

Uniqueness of static vacuum asymptotically flat black holes and equipotential photon surfaces in $n+1$ dimensions \`a la Robinson

In this paper, we combine and generalize to higher dimensions the approaches to proving the uniqueness of connected (3+1)-dimensional static vacuum asymptotically flat black hole spacetimes by M\"uller zum Hagen--Robinson--Seifert and by Robinson. Applying these techniques, we prove and/or reprove geometric inequalities for connected (n + 1)-dimensional static vacuum asymptotically flat spacetimes with either black hole or equipotential photon surface or specifically photon sphere inner boundary. In particular, assuming a natural upper bound on the total scalar curvature of the boundary, we recover and extend the well-known uniqueness results for such black hole and equipotential photon surface spacetimes. We also relate our results and proofs to existing results, in particular to those by Agostiniani--Mazzieri and by Nozawa--Shiromizu--Izumi--Yamada.

math.DG↗
arXiv2023

Mass and topology of a static stellar model

This study investigates the topological implications arising from stable (free boundary) minimal surfaces in a static perfect fluid space while ensuring that the fluid satisfies certain energy conditions. Based on the main findings, it has been established the topology of the level set $\{f=c\}$ (the boundary of a stellar model), where $c$ is a positive constant and $f$ is the static potential of a static perfect fluid space. We prove a non-existence result of stable free boundary minimal surfaces in a static perfect fluid space. An upper bound for the Hawking mass for the level set $\{f=c\}$ in a non-compact static perfect fluid space was derived, and the positivity of Hawking mass is provided in the compact case when the boundary $\{f=c\}$ is a topological sphere. We dedicate a section to revisit the Tolman-Oppenheimer-Volkoff solution, an important procedure for producing static stellar models. We will present a new static stellar model inspired by Witten's black hole (or Hamilton's cigar).

math.DG↗
arXiv2022

Minimal surfaces and CPE metric

The critical points of the total scalar curvature functional, restricted to closed $n$-dimensional manifolds with constant scalar curvature metrics and unit volume, are termed CPE metrics. In 1987, Arthur L. Besse conjectured that CPE metrics are always Einstein. Using the theory of minimal surfaces, we prove the conjecture for three-dimensional manifolds with $C^\infty$-generic Riemannian metric.

math.DG↗
arXiv2022

Electrostatic system with divergence-free Bach tensor and non-null cosmological constant

We prove that three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat, provide that the electric field and the gradient of the lapse function are linearly dependent. Consequently, a three-dimensional electrostatic manifold admits a local warped product structure with a one-dimensional base and a constant curvature surface fiber.

math.DG↗
arXiv2022

Rigidity results for shrinking and expanding Ricci solitons

In this paper, we prove some rigidity results for both shrinking and expanding Ricci solitons. First, we prove that compact shrinking Ricci solitons are Einstein if we control the maximum value of the potential function. Then, we prove some rigidity results for non-compact gradient expanding and shrinking Ricci solitons with pinched Ricci and scalar curvatures, assuming an asymptotic condition on the scalar curvature at infinity.

math.DG↗
arXiv2022

A comparison theorem for steady Ricci solitons

We prove that a steady gradient Ricci soliton is either Ricci flat with a constant potential function or a quotient of the product steady soliton $N^{n-1}\times\mathbb{R}$, where $N^{n-1}$ is Ricci flat, or isometric to the Bryant soliton (up to scalings), provided that a couple of geometric conditions inspired by the cigar soliton hold. As an application, we prove that any complete non-compact steady Ricci soliton with positive Ricci curvature controlled by the scalar curvature $R$, curvature tensor $Rm$ satisfying $|Rm|r\to o(1)$ and $R\to\infty$, as $r\to\infty$, must be the Bryant soliton. Moreover, we prove that any complete steady soliton with positively pinched Ricci curvature must be Ricci flat.

math.DG↗
arXiv2022

Mean-stable surfaces in Static Einstein-Maxwell theory

In this paper we use the theory of mean-stable surfaces (stable minimal surfaces included) to explore the static Einstein-Maxwell space-time. We first prove that the zero set of the lapse function must be contained in the horizon boundary. Then, we explore some implications of it providing some results of nonexistence of stable minimal surfaces in the interior of an electrostatic space, subject to a certain initial boundary data. We finish by proving that the ADM mass is bounded from above by the Hawking quasi-local mass.

math.DG↗
arXiv2021

Self-similar solutions to the mean curvature flow in $\mathbb{R}^{3}$

In this paper we make an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in $\mathbb{R}^{3}$. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in $\mathbb{R}^{3}$ are trivial. Moreover, we characterize the self-similar solutions of the MCF by surfaces of revolutions under a homothetic helicoidal motion in $\mathbb{R}^{3}$ in terms of the curvature of the generating curve. Finally, we characterize the self-similar solutions for the MCF by cylindrical surfaces under a homothetic helicoidal motion in $\mathbb{R}^3$. Explicit families of exact solutions for the MCF by cylindrical surfaces in $\mathbb{R}^{3}$ are also given.

math.DG↗
arXiv2021

On the geometry of electrovacuum spaces in higher dimensions

A classical question in general relativity is about the classification of regular static black hole solutions of the static Einstein-Maxwell equations (or electrovacuum system). In this paper, we prove some classification results for an electrovacuum system such that the electric potential is a smooth function of the lapse function. In particular, we show that an n-dimensional locally conformally flat extremal electrovacuum space must be in the Majumdar-Papapetrou class. Also, we prove that any three or four dimensional extremal electrovacuum space must be locally conformally flat. Moreover, we prove that an n-dimensional subextremal electrovacuum space with fourth-order divergence free Weyl tensor and zero radial Weyl curvature such that the electric potential is in the Reissner-Nordström class is locally a warped product manifold with (n-1)-dimensional Einstein fibers. Finally, a three dimensional subextremal electrovacuum space with third-order divergence free Cotton tensor was also classified.

gr-qc↗
arXiv2021

Static Einstein-Maxwell space-time invariant by translation

In this paper we study the static Einstein-Maxwell space when it is conformal to an $n$-dimensional pseudo-Euclidean space, which is invariant under the action of an $(n-1)$-dimensional translation group. We also provide a complete classification of such space.

gr-qc↗
arXiv2021

Vanishing conditions on Weyl tensor for Einstein-type manifolds

In this paper we consider an Einstein-type equation which generalizes important geometric equations, like static and critical point equations. We prove that a complete Einstein-type manifold with fourth-order divergence-free Weyl tensor and zero radial Weyl curvature is locally a warped product with $(n-1)$-dimensional Einstein fibers, provided that the potential function is proper. As a consequence, we prove a result about the nonexistence of multiple black holes in static spacetimes.

math.DG↗
arXiv2021

On the fluid ball conjecture

The fluid ball conjecture states that a static perfect fluid space-time is spherically symmetric. In this paper we construct a Robinson's divergence formula for the static perfect fluid space-time. Inspired by this conjecture, a rigidity result for the spatial factor of a static perfect fluid satisfying some boundary conditions is proved, provided that an equation of state holds.

math.DG↗
arXiv2020

An analysis of symmetry groups of generalized $m$-quasi-Einstein manifolds

In this paper emphasis is placed on how the behavior of the solutions of a PDE is affected by the geometry of the generalized $m$-quasi-Einstein manifold, and vice versa. Considering a $n$-dimensional generalized $m$-quasi-Einstein manifold which is conformal to a pseudo-Euclidean space, we prove the most general symmetry group of maximal dimension. Moreover, we demonstrate that there is no different low dimensional invariant on a generalized $m$-quasi-Einstein manifold. As an application, we use the invariant structure of the metric to provide an example of shrinking $m$-quasi-Einstein manifold (cf. Example 3). A discussion about the fluid ball conjecture was made.

math.DG↗