arXiv subjects

Ezequiel Barbosa

Publications and source records attributed to Ezequiel Barbosa.

At least 19 recordsLinked to original sources

arXiv2026

Anisotropic Obstacle Problems for Minimal Surfaces: Regularity of the Free Boundary via the Cahn-Hoffman Transform

We study an obstacle problem for surfaces minimizing an anisotropic surface energy of ellipsoidal type. Given a convex obstacle and a boundary datum, we seek a surface that minimizes the anisotropic area functional while remaining above the obstacle. The central novelty is the systematic use of the Cahn-Hoffman transform to convert the anisotropic problem into an equivalent isotropic problem with a generalized Robin boundary condition. We prove optimal regularity of the solution ($C^{1,1}$ up to the free boundary) and $C^{1,α}$-regularity of the free boundary itself under a non-degeneracy condition. The singular set of the free boundary is shown to have Hausdorff dimension at most $n-1$, and a logarithmic epiperimetric inequality yields its $(n-1)$-rectifiability. The approach combines Caffarelli's classical theory of obstacle problems with the geometric theory of anisotropic mean curvature and the Alexandrov reflection principle adapted to the anisotropic setting.

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arXiv2026

Anisotropic Parabolic Obstacle Problems and the Stefan Problem: Regularity of the Evolving Free Boundary

We study a parabolic obstacle problem for surfaces evolving by anisotropic mean curvature flow subject to an obstacle constraint. Given a convex obstacle and initial data, we seek an evolving surface minimizing an anisotropic energy functional while remaining above the obstacle; as a special case, this framework includes the anisotropic Stefan problem, where the free boundary represents a phase transition interface with direction-dependent surface tension. The central tool is the Cahn--Hoffman transform $S(x) = A^{-1/2}x$, which maps the Wulff ellipsoid $\{x : x^T A^{-1}x \leq 1\}$ to the Euclidean unit ball and converts the anisotropic problem into an equivalent isotropic one with a generalized Robin-type condition on the free boundary. We prove optimal regularity of the solution ($C^{1,α}$ in space and $C^{0,α/2}$ in time up to the free boundary) and $C^{1,α}$-regularity of the evolving free boundary at non-degenerate points. The parabolic Hausdorff dimension of the space-time singular set is shown to be at most $n - 1$.

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arXiv2024

Operator $Δ-aS$ on warped product manifolds

In this work we studied the stability of the family of operators $L_a=Δ-aS$, $a\in\mathbb R$, in a warped product of an infinite interval or real line by one compact manifold, where $Δ$ is the Laplacian and $S$ is the scalar curvature of the resulting manifold.

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arXiv2023

Area rigidity for the equatorial disk in the ball

It is proved by Brendle in [4] that the equatorial disk $D^k$ has least area among $k$-dimensional free boundary minimal surfaces in the Euclidean ball $B^n$. By comparing the excess of free boundary minimal surfaces with the excess of the associated cones over the boundary, we prove the existence of a gap for the area.

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arXiv2023

Gap phenomena for constant mean curvature surfaces

In this paper, we prove gap results for constant mean curvature (CMC) surfaces. Firstly, we find a natural inequality for CMC surfaces which imply convexity for distance function. We then show that if $Σ$ is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then $Σ$ is either a sphere or a right circular cylinder. Next, we show that if $Σ$ is a free boundary CMC surface in the Euclidean 3-ball satisfying the same inequality, then either $Σ$ is a totally umbilical disk or an annulus of revolution. These results complete the picture about gap theorems for CMC surfaces in the Euclidean 3-space. We also prove similar results in the hyperbolic space and in the upper hemisphere, and in higher dimensions.

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arXiv2022

Proper free-boundary minimal hypersurfaces with a rotational symmetry in the Schwarzschild space

In this work we present a new family of properly embedded free boundary minimal hypersurfaces of revolution with circular boundaries in the horizon of the $n$-dimensional Schwarzschild space, $n\geq3$. In particular, we answer a question proposed by O. Chodosh and D. Ketover \cite{CK} on the existence of non-totally geodesic minimal surfaces in the 3-dimensional Schwarzschild space.

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arXiv2022

Uniqueness of free-boundary minimal hypersurfaces in rotational domains

In this work, we investigate the existence of compact free-boundary minimal hypersurfaces immersed in several domains. Using an original integral identity for compact free-boundary minimal hypersurfaces that are immersed in a domain whose boundary is a regular level set, we study the case where this domain is a quadric or, more generally, a rotational domain. This existence study is done without topological restrictions. We also obtain a new gap theorem for free boundary hypersurfaces immersed in an Euclidean ball and in a rotational ellipsoid.

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arXiv2022

Some Properties of the Intersection of Free Boundary Minimal Hypersurfaces in Euclidean Balls

In this work, we prove that any two free boundary minimal hypersurfaces in the unit Euclidean ball have an intersection point in any half-ball. This is a strong version of the Frankel property proved by A. Fraser and M. Li \cite{FRLI}. As a consequence, we obtain the two-piece property for free boundary minimal hypersurfaces in the unit ball: every equatorial disk divides any compact minimal hypersurface with free boundary in the unit ball in two connected pieces.

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arXiv2021

Disks area-minimizing in mean convex Riemannian $n$-manifolds

We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed disks whose boundaries are homotopically non-trivial curves in an oriented compact manifold which possesses convex mean curvature boundary, positive escalar curvature and admits a map to $\mathbb{D}^2\times T^{n}$ with nonzero degree, where $\mathbb{D}^2$ is a disk and $T^n$ is an $n$-dimensional torus. We also prove a rigidity result for the equality case when the boundary is totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in \cite{AMB} to higher dimensions.

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arXiv2021

On free boundary minimal hypersurfaces in the Riemannian Schwarzschild space

In contrast with the 3-dimensional case (cf. \cite{RaMo}), where rotationally symmetric totally geodesic free boundary minimal surfaces have Morse index one; we prove in this work that the Morse index of a free boundary rotationally symmetric totally geodesic hypersurface of the $n$-dimensional Riemannnian Schwarzschild space with respect to variations that are tangential along the horizon is zero, for $n\geq4$. Moreover, we show that there exist non-compact free boundary minimal hypersurfaces which are not totally geodesic, $n\geq 8$, with Morse index equal to $0$. Also, it is shown that, for $n\geq4$, there exist infinitely many non-compact free boundary minimal hypersurfaces, which are not congruent to each other, with infinite Morse index. We also study the density at infinity of a free boundary minimal hypersurface with respect to a minimal cone constructed over a minimal hypersurface of the unit Euclidean sphere. We obtain a lower bound for the density in terms of the area of the boundary of the hypersurface and the area of the minimal hypersurface in the unit sphere. This lower bound is optimal in the sense that only minimal cones achieve it.

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arXiv2021

Uniqueness for the Brezis-Nirenberg type problems on spheres and hemispheres

In this work, we develop a study involving some nonlinear partial differential equations on spheres and hemispheres, with the zero Neumann boundary condition, which are so-called Brezis-Nirenberg type problems, and we give conditions on which such equations have only constant solutions. We also extend these results for some nonlinear partial differential systems.

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arXiv2021

Infinitely many sign-changing solutions of a critical fractional equation

In this paper, we obtain nonexistence results of positive solutions, and also the existence of an unbounded sequence of solutions that changing sign for some critical problems involving conformally invariant operators on the standard unit sphere, and the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together to blow up arguments, a variant of Pohozaev's type identity, a refinement of regularity results for this type operators, and finally, by exploiting the symmetries of the sphere.

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arXiv2020

Gap results for free boundary CMC surfaces in conformally Euclidean three-balls

In this work, we consider $M=(\mathbb{B}^3_r,\bar{g})$ as the Euclidean three-ball with radius $r$ equipped with the metric $\bar{g}=e^{2h}\left\langle , \right\rangle$ conformal to the Euclidean metric. We show that if a free boundary CMC surface $Σ$ in $M$ satisfies a pinching condition on the length of the traceless second fundamental tensor which involves the support function of $Σ$, the positional conformal vector field $\vec{x}$ and its potential function $σ,$ then either $Σ$ is a disk or $Σ$ is an annulus rotationally symmetric. In a particular case, we construct an example of minimal surface with strictly convex boundary in $M$, when $M$ is the Gaussian space, that illustrate our results. These results extend to the CMC case and to many others different conformally Euclidean spaces the main result obtained by Haizhong Li and Changwei Xiong.

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arXiv2020

A characterization of the delaunay surfaces

In this paper we use the Alexandrov Reflection Method to obtain a characterization to embedded CMC capillary annulus $Σ^2 \subset \mathbb{B}^3$. In especial, but using a new strategy, we present a new characterization to the critical catenoid. Precisely, we show that $Σ\subset \mathbb{B}^3$ being an embedded minimal free boundary annulus in $\mathbb{B}^3$ such that $\partial Σ$ is invariant under reflection through a coordinates planes, then $Σ$ is the critical catenoid. This work is part of the second author thesis which was written in 2019.

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arXiv2019

Topological obstructions to nonnegative scalar curvature and mean convex boundary

We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex boundary. For example, we show that the manifold $(T^{n-2}\times Σ)\# N$, where $Σ$ is a compact, connected and orientable surface which is not a disk or a cylinder and $N$ is a closed $n$-dimensional manifold, does not admit a metric of non-negative scalar curvature and mean convex boundary, and the manifold $(I\times T^{n-1})\#N$, where $I=[a,b]$, does not admit a metric of positive scalar curvature and mean convex boundary.

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arXiv2018

Min-oo conjecture for fully nonlinear conformally invariant equations

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with. This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$, where $D(r)$ denotes a geodesic ball of radius $r\in (0,π/2]$ in $\mathbb S^n$, and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, $(M,g)$ must be isometric to the closed geodesic ball $\overline{D(r)}$. As a side product, in dimension $2$ our methods provide a new proof to Toponogov's Theorem about the rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's Theorem is equivalent to a rigidity theorem for spherical caps in the Hyperbolic three-space $\mathbb H^3$. In fact, we extend it to obtain rigidity for super-solutions to certain Monge-Ampère equations.

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arXiv2018

A remark on a curvature gap for minimal surfaces in the ball

We extend to higher codimension earlier characterization of the equatorial disk and the critical catenoid by a pinching condition on the length of their second fundamental form among free boundary minimal surfaces in the three dimensional Euclidean ball due to L. Ambrozio and I. Nunes.

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arXiv2018

Uniqueness results for free-boundary minimal hypersurfaces in conformally Euclidean balls and annular domains

In this paper we prove that a flat free-boundary minimal $n$-disk, $n\geq3$, in the unit Euclidean ball $B^{n+1}$ is the unique compact free boundary minimal hypersurface in the unit Euclidean ball which the squared norm of the second fundamental form is less than either $\frac{n^2}{4}$ or $\frac{(n-2)^2}{4|x|^2}$. Moreover, we prove analogous results for compact free boundary minimal hypersurfaces in annular domains with a conformally Euclidean metric.

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