arXiv subjects

Abraão Mendes

Publications and source records attributed to Abraão Mendes.

At least 19 recordsLinked to original sources

arXiv2026

Second Jacobi eigenvalues and spectral rigidity for surfaces in Berger spheres

In this paper, we establish an upper bound for the second eigenvalue of the scalar Jacobi operator of an arbitrary closed two-sided surface immersed in a contracted Berger sphere. The estimate involves the total squared mean curvature, the Euler characteristic, and an explicit nonpositive term determined by the angle between the surface normal and the Hopf direction. The proof combines the realization of a Berger sphere as a geodesic hypersurface of a complex projective plane, the first standard embedding of the latter into a Euclidean sphere, a weighted Hersch--Li--Yau balancing argument, and the conformal invariance of the Willmore functional. No minimality or constant mean curvature assumption is imposed. As an application, if $1/3\leqα\leq1$ and the surface has nonpositive Euler characteristic, then its second Jacobi eigenvalue is nonpositive; it is strictly negative for $α>1/3$. At the critical value $α=1/3$, equality forces the immersed image to be congruent to the minimal Clifford torus; in the embedded category, this yields a complete characterization of the equality case.

math.DG↗

arXiv2026

A strict second-eigenvalue estimate for the scalar stability operator on surfaces in $\mathbb{RP}^{3}$

Let $φ:Σ^2\looparrowright\mathbb{RP}^3$ be a closed immersed surface, with no orientability or, equivalently, two-sidedness assumptions. We establish the sharp quantitative estimate $$ λ_2(Δ+|σ|^2+2)\leq2-\frac{2}{\operatorname{Area}(Σ)}\int_ΣH^2dΣ+\frac{4πχ(Σ)}{\operatorname{Area}(Σ)}. $$ Our main contribution is the analysis of the borderline case, which rules out equality when $χ(Σ)\leq0$. More precisely, $$ λ_2(Δ+|σ|^2+2)<2\quad\text{whenever}\quadχ(Σ)\leq0. $$ The proof combines the canonical Veronese embedding of $\mathbb{RP}^3$ into $\mathbb{S}^8$, the conformal test function method, an Obata-type rigidity argument, and the classification of closed flat minimal surfaces in $\mathbb{RP}^3$.

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arXiv2026

Second Eigenvalue Estimates and Spectral Rigidity for Submanifolds of Compact Rank-One Symmetric Spaces

In this paper, we establish upper bounds for the second eigenvalue of the Schrödinger operator $L=Δ+|σ|^2+k$ on $k$-dimensional closed submanifolds of the compact projective spaces $\mathbb F P^m$, where $\mathbb F\in\{\mathbb R,\mathbb C,\mathbb H\}$, as well as the Cayley projective plane. The estimates are obtained by combining the standard spherical embeddings of these spaces with a conformal test-function argument. In the complex, quaternionic, and Cayley cases, the resulting bounds involve correction terms that record the position of the tangent spaces of the submanifold relative to the corresponding geometric structures. We also derive universal estimates depending only on the dimension of the submanifold and the underlying division algebra. We show that equality in the sharp estimates forces the submanifold to be totally umbilical. Finally, using known classifications of totally umbilical submanifolds, we compute the second eigenvalue for the standard real, complex, quaternionic, and Cayley models and identify the totally geodesic examples that attain the sharp upper bounds.

math.DG↗

arXiv2026

Index estimates for planar domains with Robin boundary condition

In this note, we study the index of the Laplace operator on planar domains $Ω$ with compact smooth boundary, allowing both the compact and noncompact cases, under a natural Robin boundary condition involving the geodesic curvature of $\partialΩ$. We obtain lower bounds for the index in terms of the number of boundary components. Our approach combines conformal and spectral techniques with the topology of the domain, encoded in the space of $L^2$ harmonic vector fields tangent to the boundary. The result is motivated by index estimates for free boundary minimal surfaces and provides an intrinsic counterpart to a borderline geometric situation arising in that setting.

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arXiv2025

Stability and rigidity of axisymmetric marginally outer trapped two-spheres

In [7], H. Bray, S. Brendle, and A. Neves studied rigidity properties of area-minimizing two-spheres in Riemannian three-manifolds with uniformly positive scalar curvature. In [13], these results were extended to marginally outer trapped surfaces (MOTS) in general initial data sets $(M^3,g,K)$ under a natural energy condition. In the present work, we refine the latter results to the setting of axisymmetric MOTS in initial data sets admitting a nontrivial Killing vector field. Conditions for the stability of such MOTS, as well as a new foliation lemma by axisymmetric surfaces of constant outward null expansion, are obtained. Finally, we discuss some aspects of the rotating Nariai spacetimes and their relation to these results.

math.DG↗

arXiv2025

Area-charge inequalities and rigidity of time-symmetric initial data sets

In this paper, we establish new area-charge inequalities for the boundary of time-symmetric Einstein-Maxwell initial data sets, in both compact and noncompact cases, under the dominant energy condition. These inequalities lead to novel rigidity theorems with no analogues in the uncharged setting. In the noncompact case, our result is obtained by applying Gromov's $μ$-bubble technique in a new geometric context.

gr-qc↗

arXiv2025

Rigidity of surfaces with nonpositive Euler characteristic by the second eigenvalue of the Jacobi operator

In this paper, we investigate the spectral properties of the Jacobi operator for immersed surfaces with nonpositive Euler characteristic, extending previous results in the field. We first prove a sharp upper bound for the second eigenvalue of the Jacobi operator for compact surfaces with nonpositive Euler characteristic that are fully immersed in the Euclidean sphere, and then we classify all such surfaces attaining this upper bound. Furthermore, we demonstrate that totally geodesic tori maximize the second eigenvalue among all compact orientable surfaces with positive genus in the product space $\mathbb{S}^1(r) \times \mathbb{S}^2(s)$.

math.DG↗

arXiv2025

Area-charge inequality and local rigidity in charged initial data sets

This paper investigates the geometric consequences of equality in area-charge inequalities for spherical minimal surfaces and, more generally, for marginally outer trapped surfaces (MOTS), within the framework of the Einstein-Maxwell equations. We show that, under appropriate energy and curvature conditions, saturation of the inequality $\mathcal{A} \geq 4π(\mathcal{Q}_{\rm E}^2 + \mathcal{Q}_{\rm M}^2)$ imposes a rigid geometric structure in a neighborhood of the surface. In particular, the electric and magnetic fields must be normal to the foliation, and the local geometry is isometric to a Riemannian product. We establish two main rigidity theorems: one in the time-symmetric case and another for initial data sets that are not necessarily time-symmetric. In both cases, equality in the area-charge bound leads to a precise characterization of the intrinsic and extrinsic geometry of the initial data near the critical surface.

math.DG↗

arXiv2025

Rigidity results for free boundary hypersurfaces in initial data sets with boundary

In this work, we present several rigidity results for compact free boundary hypersurfaces in initial data sets with boundary. Specifically, in the first part of the paper, we extend the local splitting theorems from [G. J. Galloway and H. C. Jang, Some scalar curvature warped product splitting theorems, Proc. Am. Math. Soc. 148 (2020), no. 6, 2617-2629] to the setting of manifolds with boundary. To achieve this, we build on the approach of the original paper, utilizing results on free boundary marginally outer trapped surfaces (MOTS) applied to specific initial data sets. In the second part, we extend the main results from [A. Barros and C. Cruz, Free boundary hypersurfaces with non-positive Yamabe invariant in mean convex manifolds, J. Geom. Anal. 30 (2020), no. 4, 3542-3562] to the context of free boundary MOTS in initial data sets with boundary.

math.DG↗

arXiv2024

Aspects of the geometry and topology of expanding horizons

The aim of this paper is to extend some basic results about marginally outer trapped surfaces to the context of surfaces having general null expansion. Motivated in part by recent work of Chai-Wan, we introduce the notion of $\mathfrak{g}$-stability for a general closed hypersurface $Σ$ in an ambient initial data set and prove that, under natural energy conditions, $Σ$ has positive Yamabe type, that is, $Σ$ admits a metric of positive scalar curvature, provided $Σ$ is $\mathfrak{g}$-stable. A similar result is obtained when $Σ$ is embedded in a null hypersurface of a spacetime satisfying the dominant energy condition. Area bounds under similar conditions are obtained in the case where $Σ$ is $2$-dimensional. Conditions implying $\mathfrak{g}$-stability are also discussed. Finally, we obtain a spacetime positive mass theorem for initial data sets with compact boundary $Σ$ of positive null expansion, assuming that the dominant energy condition is sufficiently strict near $Σ$. This extends recent results of Galloway-Lee and Lee-Lesourd-Unger.

gr-qc↗

arXiv2024

Some rigidity results for charged initial data sets

In this note, we consider some initial data rigidity results concerning marginally outer trapped surfaces (MOTS). As is well known, MOTS play an important role in the theory of black holes and, at the same time, are interesting spacetime analogues of minimal surfaces in Riemannian geometry. The main results presented here expand upon earlier works by the authors, specifically addressing initial data sets incorporating charge.

math.DG↗

arXiv2023

Rigidity of min-max minimal disks in $3$-balls with non-negative Ricci curvature

In this paper we prove a rigidity statement for free boundary minimal surfaces produced via min-max methods. More precisely, we prove that for any Riemannian metric $g$ on the 3-ball $B$ with non-negative Ricci curvature and $\mathrm{II}_{\partial B}\ge g_{|\partial B}$, there exists a free boundary minimal disk $Δ$ of least area among all free boundary minimal disks in $(B,g)$. Moreover, the area of any such $Δ$ equals to the width of $(B,g)$, $Δ$ has index one, and the length of $\partialΔ$ is bounded from above by $2π$. Furthermore, the length of $\partialΔ$ equals to $2π$ if and only if $(B,g)$ is isometric to the Euclidean unit ball. This is related to a rigidity result obtained by F.C. Marques and A. Neves in the closed case. The proof uses a rigidity statement concerning half-balls with non-negative Ricci curvature which is true in any dimension.

math.DG↗

arXiv2023

Initial data rigidity results

We present several rigidity results for initial data sets motivated by the positive mass theorem. An important step in our proofs here is to establish conditions that ensure that a marginally outer trapped surface is "weakly outermost". A rigidity result for Riemannian manifolds with a lower bound on their scalar curvature is included as a special case.

gr-qc↗

arXiv2023

Some rigidity results for compact initial data sets

In this paper, we prove several rigidity results for compact initial data sets, in both the boundary and no boundary cases. In particular, under natural energy, boundary, and topological conditions, we obtain a global version of the main result in [15]. We also obtain some extensions of results in [12]. A number of examples are given in order to illustrate some of the results presented in this paper.

gr-qc↗

arXiv2022

Free boundary minimal hypersurfaces outside of the ball

In this paper we obtain two classification theorems for free boundary minimal hypersurfaces outside of the unit ball (exterior FBMH for short) in Euclidean space. The first result states that the only exterior stable FBMH with parallel embedded regular ends are the catenoidal hypersurfaces. To achieve this we prove a Bôcher type result for positive Jacobi functions on regular minimal ends in $\mathbb{R}^{n+1}$ which, after some calculations, implies the first theorem. The second theorem states that any exterior FBMH $Σ$ with one regular end is a catenoidal hypersurface. Its proof is based on a symmetrization procedure similar to R. Schoen [14]. We also give a complete description of the catenoidal hypersurfaces, including the calculation of their indices.

math.DG↗

arXiv2021

Rigidity of free boundary MOTS

The aim of this work is to present an initial data version of Hawking's theorem on the topology of back hole spacetimes in the context of manifolds with boundary. More precisely, we generalize the results of G. J. Galloway and R. Schoen [13] and G. J. Galloway [11, 12] by proving that a compact free boundary stable marginally outer trapped surface (MOTS) $Σ$ in an initial data set with boundary satisfying natural dominant energy conditions (DEC) is of positive Yamabe type, i.e. $Σ$ admits a metric of positive scalar curvature with minimal boundary, provided $Σ$ is outermost. To do so, we prove that if $Σ$ is a compact free boundary stable MOTS which does not admit a metric of positive scalar curvature with minimal boundary in an initial data set satisfying the interior and the boundary DEC, then an outer neighborhood of $Σ$ can be foliated by free boundary MOTS $Σ_t$, assuming that $Σ$ is weakly outermost. Moreover, each $Σ_t$ has vanishing outward null second fundamental form, is Ricci flat with totally geodesic boundary, and the dominant energy conditions saturate on $Σ_t$.

math.DG↗

arXiv2018

Vanishing theorems for the cohomology groups of free boundary hypersurfaces

In this paper, we prove that there exists a universal constant $C$, depending only on positive integers $n\geq 3$ and $p\leq n-1$, such that if $M^n$ is a compact free boundary submanifold of dimension $n$ immersed in the Euclidean unit ball $\mathbb{B}^{n+k}$ whose size of the traceless second fundamental form is less than $C$, then the $p$th cohomology group of $M^n$ vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball $\mathbb{B}^{2+k}$.

math.DG↗

arXiv2018

Characterization of hypersurfaces via the second eigenvalue of the Jacobi operator

In this work we characterize certain immersed closed hypersurfaces of some ambient manifolds via the second eigenvalue of the Jacobi operator. First, we characterize the Clifford torus as the surface which maximizes the second eigenvalue of the Jacobi operator among all closed immersed orientable surfaces of $\mathbb S^3$ with genus bigger than zero. After, we characterize the slices of the warped product $I\times_h\mathbb S^n$, under a suitable hypothesis on the warping function $h:I\subset\mathbb R\to\mathbb R$, as the only hypersurfaces which saturate a certain integral inequality involving the second eigenvalue of the Jacobi operator. As a consequence, we obtain that if $Σ$ is a closed immersed hypersurface of $\mathbb R\times\mathbb S^n$, then the second eigenvalue of the Jacobi operator of $Σ$ satisfies $λ_2\le n$ and the slices are the only hypersurfaces which satisfy $λ_2=n$.

math.DG↗