arXiv subjects

Ilton Menezes

Publications and source records attributed to Ilton Menezes.

5 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↗

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

Rigity results on $ρ$-Einstein solitons with zero scalar curvature

In this paper we show that a $ρ$-Einstein solitons conformal to a pseudo-Euclidean space, invariant under the action of the pseudo-orthogonal group with zero scalar curvature is stady and consequently flat. How application of the results obtained we present an explicit example for a the question proposed by Kazdan in [17].

math.DG↗

arXiv2020

Rigidity results on gradient Schouten solitons

In this paper we consider $\rho$-Einstein solitons of type $M= \left(B^n, g^{*}\right) \times (F^m,g_F)$, where $\left(B^n,g^{*}\right)$ is conformal to a pseudo-Euclidean space and invariant under the action of the pseudo-orthogonal group, and $\left(F^m,g_{F}\right)$ is an Einstein manifold. We provide all the solutions for the gradient Schouten soliton case. Moreover, in the Riemannian case, we prove that if $M= \left(B^n, g^{*}\right) \times (F^m,g_F)$ is a complete gradient Schouten soliton then $\left(B^{n},g^{*}\right)$ is isometric to $\mathbb{S}^{n-1}\times \mathbb{R}$ and $F^m$ is a compact Einstein manifold.

math.DG↗