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Frederico Girão

Publications and source records attributed to Frederico Girão.

15 recordsLinked to original sources
arXiv2025

Mass from an Extrinsic Point of View

We express the $q$-th Gauss-Bonnet-Chern mass of an immersed submanifold of Euclidean space as a linear combination of two terms: the total $(2q)$-th mean curvature and the integral, over the entire manifold, of the inner product between the $(2q+1)$-th mean curvature vector and the position vector of the immersion. As a consequence, we obtain, for each $q$, a geometric inequality that holds whenever the positive mass theorem (for the $q$-th Gauss-Bonnet-Chern mass) holds.

math.DG↗
arXiv2021

Extrinsic black hole uniqueness in pure Lovelock gravity

We define a notion of extrinsic black hole in pure Lovelock gravity of degree $k$ which captures the essential features of the so-called Lovelock-Schwarzschild solutions, viewed as rotationally invariant hypersurfaces with null $2k$-mean curvature in Euclidean space $\mathbb R^{n+1}$, $2\leq 2k\leq n-1$. We then combine a regularity argument with a rigidity result by Araújo-Leite to prove, under a natural ellipticity condition, a global uniqueness theorem for this class of black holes. As a consequence we obtain, in the context of the corresponding Penrose inequality for graphs established by Ge-Wang-Wu, a local rigidity result for the Lovelock-Schwarzschild solutions.

math.DG↗
arXiv2021

A rigidity result for the graph case of the Penrose inequality

In this note we prove a global rigidity result for asymptotically flat, scalar flat Euclidean hypersurfaces with a minimal horizon lying in a hyperplane, under a natural ellipticity condition. As a consequence we obtain, in the context of the Riemannian Penrose conjecture, a local rigidity result for the family of exterior Schwarzschild solutions (viewed as graphs in Euclidean space).

math.DG↗
arXiv2021

A Penrose inequality for asymptotically locally hyperbolic graphs

We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for a class of hypersurfaces in certain locally hyperbolic manifolds. As an application we derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension $n\geq 3$. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chruściel and Simon on the validity of a Penrose-type inequality for black hole solutions carrying a higher genus horizon.

math.DG↗
arXiv2021

An Alexandrov-Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality

We prove a sharp Alexandrov-Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic $n$-space, $n\geq 3$. The argument uses two new monotone quantities along the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter-Schwarzschild solution. This sharpens previous results by Dahl-Gicquaud-Sakovich and settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space-times with negative cosmological constant. We also explain how our methods can be easily adapted to derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension $n\geq 3$. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chrusciel and Simon on the validity of a Penrose-type inequality for exotic black holes.

math.DG↗
arXiv2019

Weighted Alexandrov-Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang and Wu

We consider a conjecture made by Ge, Wang and Wu regarding weighted Alexandrov-Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the $k^{\mathrm{th}}$ mean curvature in terms of the area of the hypersurface). We prove an inequality very similar to the conjectured one. Moreover, when $k$ is zero and the ambient space has dimension three, we give a counterexample to the conjectured inequality.

math.DG↗
arXiv2019

Weighted geometric inequalities for hypersurfaces in sub-static manifolds

We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region enclosed by the hypersurface. The second one involves the total weighted mean curvature and the area of the hypersurface. Versions of the first inequality for the sphere and for the adS-Reissner-Nordström manifold are proven. We end with an example of a convex surface for which the ratio between the polar moment of inertia and the square of the area is less than that of the round sphere.

math.DG↗
arXiv2018

The mass in terms of Einstein and Newton

It is shown that the mass of an asymptotically flat manifold with a noncompact boundary can be computed in terms of limiting surface integrals involving the Einstein tensor of the interior metric and the Newton tensor attached to the second fundamental form of the boundary. This extends to this setting previous results by several authors in the boundaryless case. The method outlined below, which is based on a coordinate-free approach due to Herzlich, also applies to asymptotically hyperbolic manifolds, again with a noncompact boundary, for which a similar notion of mass has been recently considered by Almaraz and the first named author, and both cases will be discussed here.

math.DG↗
arXiv2014

On the limiting behavior of the Brown-York quasi-local mass in asymptotically hyperbolic manifolds

We show that the limit at infinity of the vector-valued Brown-York-type quasi-local mass along any coordinate exhaustion of an asymptotically hyperbolic $3$-manifold satisfying the relevant energy condition on the scalar curvature has the conjectured causal character. Our proof uses spinors and relies on a Witten-type formula expressing the asymptotic limit of this quasi-local mass as a bulk integral which manifestly has the right sign under the above assumptions. In the spirit of recent work by Hijazi, Montiel and Raulot, we also provide another proof of this result which uses the theory of boundary value problems for Dirac operators on compact domains to show that a certain quasi-local mass, which converges to the Brown-York mass in the asymptotic limit, has the expected causal character under suitable geometric assumptions.

math.DG↗
arXiv2014

Penrose inequalities and a positive mass theorem for charged black holes in higher dimension

We use the inverse mean curvature flow to establish Penrose-type inequalities for time-symmetric Einstein-Maxwell initial data sets which can be suitably embedded as a hypersurface in Euclidean space $\mathbb R^{n+1}$, $n\geq 3$. In particular, we prove a positive mass theorem for this class of charged black holes. As an application we show that the conjectured upper bound for the area in terms of the mass and the charge, which in dimension $n=3$ is relevant in connection with the Cosmic Censorship Conjecture, always holds under the natural assumption that the horizon is stable as a minimal hypersurface.

math.DG↗
arXiv2012

The ADM mass of asymptotically flat hypersurfaces

We provide integral formulae for the ADM mass of asymptotically flat hypersurfaces in Riemannian manifolds with a certain warped product structure in a neighborhood of infinity, thus extending Lam's recent results on Euclidean graphs to this broader context. As applications we exhibit, in any dimension, new classes of manifolds for which versions of the Positive Mass and Riemannian Penrose inequalities hold and discuss a notion of quasi-local mass in this setting. The proof explores a novel connection between the co-vector defining the ADM mass of a hypersurface as above and the Newton tensor associated to its shape operator, which takes place in the presence of an ambient Killing field.

math.DG↗
arXiv2012

Positive mass and Penrose type inequalities for asymptotically hyperbolic hypersurfaces

We establish versions of the Positive Mass and Penrose inequalities for a class of asymptotically hyperbolic hypersurfaces. In particular, under the usual dominant energy condition, we prove in all dimensions $n\geq 3$ an optimal Penrose inequality for certain graphs in hyperbolic space $\mathbb H^{n+1}$ whose boundary has constant mean curvature $n-1$.

math.DG↗