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Ana Menezes

Publications and source records attributed to Ana Menezes.

11 recordsLinked to original sources
arXiv2023

Eigenvalue problems and free boundary minimal surfaces in spherical caps

Given a compact surface with boundary, we introduce a family of functionals on the space of its Riemannian metrics, defined via eigenvalues of a Steklov-type problem. We prove that each such functional is uniformly bounded from above, and we characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we determine that the maximizer in the case of a disk is a spherical cap of dimension two, and we prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions.

math.DG↗
arXiv2022

Slab Theorem and Halfspace Theorem for constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$

We prove that a properly embedded annular end of a surface in $\mathbb H^2\times\mathbb R$ with constant mean curvature $0<H\leq \frac{1}{2}$ can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature $0<H\leq \frac{1}{2}$ contained in $\mathbb H^2\times[0,+\infty)$ and with finite topology is necessarily a graph over a simply connected domain of $\mathbb H^2$. For the case $H=\frac{1}{2}$, the graph is entire.

math.DG↗
arXiv2021

A two-piece property for free boundary minimal hypersurfaces in the $(n+1)$-dimensional ball

We prove that every hyperplane passing through the origin in $\rr^{n+1}$ divides an embedded compact free boundary minimal hypersurface of the euclidean $(n+1)$-ball in exactly two connected hypersurfaces. We also show that if a region in the $(n+1)$-ball has mean convex boundary and contains a nullhomologous $(n-1)$-dimensional equatorial disk, then this region is a closed halfball. Our first result gives evidence to a conjecture by Fraser and Li in any dimension.

math.DG↗
arXiv2020

A two-piece property for free boundary minimal surfaces in the ball

We prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean $3$-ball in exactly two connected surfaces. We also show that if a region in the ball has mean convex boundary and contains a nullhomologous diameter, then this region is a closed halfball. Moreover, we prove the regularity at the corners of currents minimizing a partially free boundary problem by following ideas by Grüter and Simon. Our first result gives evidence to a conjecture by Fraser and Li.

math.DG↗
arXiv2020

On the asymptotic Plateau problem for area minimizing surfaces in $\mathbb{E}(-1,τ)$

We prove some existence and non-existence results for complete area minimizing surfaces in the homogeneous space $\mathbb{E}(-1,τ)$. As one of our main results, we present sufficient conditions for a curve $Γ$ in $\partial_{\infty} \mathbb{E}(-1,τ)$ to admit a solution to the asymptotic Plateau problem, in the sense that there exists a complete area minimizing surface in $\mathbb{E}(-1,τ)$ having $Γ$ as its asymptotic boundary.

math.DG↗
arXiv2019

On the characterization of minimal surfaces with finite total curvature in $\mathbb H^2\times\mathbb R$ and $\widetilde{\rm PSL}_2(\mathbb{R},τ)$

It is known that a complete immersed minimal surface with finite total curvature in $\mathbb H^2\times\mathbb R$ is proper, has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity (Hauswirth and Rosenberg, 2006; Hauswirth, Nelli, Sa Earp and Toubiana, 2015). In this paper we prove that these three properties characterize complete immersed minimal surfaces with finite total curvature in $\mathbb H^2\times\mathbb R$. As corollaries of this theorem we obtain characterizations for minimal Scherk-type graphs and horizontal catenoids in $\mathbb H^2\times\mathbb R$. We also prove that if a properly immersed minimal surface in $\widetilde{\rm PSL}_2(\mathbb{R},τ)$ has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity, then it must have finite total curvature.

math.DG↗
arXiv2014

A half-space theorem for ideal Scherk graphs in $M\times\mathbb R$

We prove a half-space theorem for an ideal Scherk graph $Σ\subset M\times\mathbb R$ over a polygonal domain $D\subset M,$ where $M$ is a Hadamard surface whose curvature is bounded above by a negative constant. More precisely, we show that a properly immersed minimal surface contained in $D\times\mathbb R$ and disjoint from $Σ$ is a translate of $Σ.$

math.DG↗
arXiv2014

The Alexandrov problem in a quotient space of $\mathbb H^2\times \mathbb R$

We prove an Alexandrov type theorem for a quotient space of $\mathbb H^2\times \mathbb R$. More precisely we classify the compact embedded surfaces with constant mean curvature in the quotient of $\mathbb H^2\times \mathbb R$ by a subgroup of isometries generated by a parabolic translation along horocycles of $\mathbb H^2$ and a vertical translation. Moreover, we construct some examples of periodic minimal surfaces in $\mathbb H^2\times\mathbb R$ and we prove a multi-valued Rado theorem for small perturbations of the helicoid in $\mathbb H^2\times\mathbb R$.

math.DG↗
arXiv2013

Periodic minimal surfaces in semidirect products

In this paper we prove existence of complete minimal surfaces in some metric semidirect products. These surfaces are similar to the doubly and singly periodic Scherk minimal surfaces in $\mathbb R^3$. In particular, we obtain these surfaces in the Heisenberg space with its canonical metric, and in Sol3 with a one-parameter family of non-isometric metrics.

math.DG↗
arXiv2013

On doubly periodic minimal surfaces in $\mathbb H^2 \times \mathbb R$ with finite total curvature in the quotient space

In this paper we develop the theory of properly immersed minimal surfaces in the quotient space $\mathbb H^2\times\mathbb R/G,$ where $G$ is a subgroup of isometries generated by a vertical translation and a horizontal isometry in $\mathbb H^2$ without fixed points. The horizontal isometry can be either a parabolic translation along horocycles in $\mathbb H^2$ or a hyperbolic translation along a geodesic in $\mathbb H^2.$ In fact, we prove that if a properly immersed minimal surface in $\mathbb H^2\times\mathbb R/G$ has finite total curvature then its total curvature is a multiple of $2π,$ and moreover, we understand the geometry of the ends. These theorems hold true more generally for properly immersed minimal surfaces in $M\times\mathbb S^1,$ where $M$ is a hyperbolic surface with finite topology whose ends are isometric to one of the ends of the above spaces $\mathbb H^2\times\mathbb R/G.$

math.DG↗