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Pablo Mira

Publications and source records attributed to Pablo Mira.

At least 37 records · Page 2Linked to original sources
arXiv2016

The geometry of stable minimal surfaces in metric Lie groups

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide a priori radius estimate which depends only on the maximum distance of points of the boundary $\partial M$ to a vertical geodesic of $X$. We also give a generalization of the classical Rado's Theorem in $\mathbb{R}^3$ to the context of compact minimal surfaces with graphical boundary over a convex horizontal domain in $X$, and we study the geometry, existence and uniqueness of this type of Plateau problem.

math.DG↗
arXiv2016

Uniqueness of immersed spheres in three-manifolds

Let $\mathcal{A}$ be a class of immersed surfaces in a three-manifold $M$, and assume that $\mathcal{A}$ is modeled by an elliptic PDE over each tangent plane. In this paper we solve the so-called Hopf uniqueness problem for the class $\mathcal{A}$ under the only mild assumption of the existence of a transitive family of candidate surfaces $\mathcal{S}\subset \mathcal{A}$. Specifically, we prove that any compact immersed surface of genus zero in the class $\mathcal{A}$ is a candidate sphere. This theorem unifies and extends many previous uniqueness results of different contexts. As an application, we settle in the affirmative a 1956 conjecture by A.D. Alexandrov on the uniqueness of immersed spheres with prescribed curvatures in $\mathbb{R}^3$.

math.DG↗
arXiv2016

Embeddedness of spheres in homogeneous three-manifolds

Let $X$ denote a metric Lie group diffeomorphic to $\mathbb{R}^3$ that admits an algebraic open book decomposition. In this paper we prove that if $Σ$ is an immersed surface in $X$ whose left invariant Gauss map is a diffeomorphism onto $\mathbb{S}^2$, then $Σ$ is an embedded sphere. As a consequence, we deduce that any constant mean curvature sphere of index one in $X$ is embedded.

math.DG↗
arXiv2015

A Hopf theorem for non-constant mean curvature and a conjecture of A.D. Alexandrov

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed mean curvature. As a consequence, we extend the classical Hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in R3.

math.DG↗
arXiv2014

Isolated singularities of graphs in warped products and Monge-Ampère equations

We study graphs of positive extrinsic curvature with a non-removable isolated singularity in 3-dimensional warped product spaces, and describe their behavior at the singularity in several natural situations. We use Monge-Ampère equations to give a classification of the surfaces in 3-dimensional space forms which are embedded around a non-removable isolated singularity and have a prescribed, real analytic, positive extrinsic curvature function at every point. Specifically, we prove that this space is in one-to-one correspondence with the space of regular, analytic, strictly convex Jordan curves in the 2-dimensional sphere $\mathbb{S^2}$.

math.DG↗
arXiv2013

Constant mean curvature spheres in homogeneous three-spheres

We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogenous metric, by proving that for each $H\in\mathbb{R}$, there exists a constant mean curvature $H$-sphere in the space that is unique up to an ambient isometry.

math.DG↗
arXiv2013

A classification of isolated singularities of elliptic Monge-Ampère equations in dimension two

Let $\mathcal{M}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Ampère equation ${\rm det} (D^2 z)=φ(x,y,z,Dz)>0$ whose graphs have a non-removable isolated singularity at the origin. We prove that $\mathcal{M}_1$ is in one-to-one correspondence with $\mathcal{M}_2\times Z_2$, where $\mathcal{M}_2$ is a suitable subset of the class of regular, real analytic strictly convex Jordan curves in $R^2$. We also describe the asymptotic behavior of solutions of the Monge-Ampère equation in the $C^k$-smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation ${\rm det} (D^2 z +\mathcal{A}(x,y,z,Dz))=φ(x,y,z,Dz)>0$.

math.AP↗
arXiv2013

The Gauss map of surfaces in ~PSL_2(R)

We define a Gauss map for surfaces in the universal cover of the Lie group PSL_2(R) endowed with a left-invariant Riemannian metric having a 4-dimensional isometry group. This Gauss map is not related to the Lie group structure. We prove that the Gauss map of a nowhere vertical surface of critical constant mean curvature is harmonic into the hyperbolic plane H^2 and we obtain a Weierstrass-type representation formula. This extends results in H^2 x R and the Heisenberg group Nil_3, and completes the proof of existence of harmonic Gauss maps for surfaces of critical constant mean curvature in any homogeneous manifold diffeomorphic to R^3 with isometry group of dimension at least 4.

math.DG↗
arXiv2013

Isoperimetric domains of large volume in homogeneous three-manifolds

Given a non-compact, simply connected homogeneous three-manifold $X$ and a sequence $\{Ω_n\}_n$ of isoperimetric domains in $X$ with volumes tending to infinity, we prove that as $n\to \infty $: 1. The radii of the $Ω_n$ tend to infinity. 2. The ratios $\{Area} (\partial Ω_n)/\{Vol}(Ω_n)$ converge to the Cheeger constant Ch$(X)$, which we also prove to be equal to $2H(X)$ where $H(X)$ is the critical mean curvature of $X$. 3. The values of the constant mean curvatures $H_n$ of the boundary surfaces $\partial Ω_n$ converge to $\frac{1}{2}\{Ch}(X)$. Furthermore, when Ch$(X)$ is positive, we prove that for $n$ large, $\partial Ω_n$ is well-approximated in a natural sense by the leaves of a certain foliation of $X$, where every leaf of the foliation is a surface of constant mean curvature $H(X)$.

math.DG↗
arXiv2011

The geometric Neumann problem for the Liouville equation

In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.

math.AP↗
arXiv2010

Surfaces of constant curvature in R^3 with isolated singularities

We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves in the 2-sphere with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R^3, i.e. compact convex surfaces of constant positive curvature with a finite number of singularities, and give applications to harmonic maps and constant mean curvature surfaces.

math.DG↗
arXiv2010

Constant mean curvature surfaces in 3-dimensional Thurston geometries

This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2 \times R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie group Sol3. We will focus on the problems of classifying compact CMC surfaces and entire CMC graphs in these spaces. A collection of important open problems of the theory is also presented.

math.DG↗
arXiv2009

Existence and uniqueness of constant mean curvature spheres in Sol_3

We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H>1/(\sqrt{3}), there exists a unique (up to left translations) immersed CMC H sphere S_H in Sol_3 (Hopf-type theorem). Moreover, this sphere S_H is embedded, and is therefore the unique (up to left translations) compact embedded CMC H surface in Sol_3 (Alexandrov-type theorem). The uniqueness parts of these results are also obtained for all real numbers H such that there exists a solution of the isoperimetric problem with mean curvature H.

math.DG↗
arXiv2009

The space of Lorentzian flat tori in anti-de Sitter 3-space

We describe the space of isometric immersions from the Lorentz plane $L^2$ into the 3-dimensional anti-de Sitter space, and solve several open problems of this context raised by M. Dajczer and K. Nomizu in 1981. We also obtain from the above result a description of the space of Lorentzian flat tori isometrically immersed in the anti de Sitter space in terms of pairs of closed curves with wave front singularities in the hyperbolic plane satisfying some compatibility conditions.

math.DG↗
arXiv2006

The Bonnet problem for surfaces in homogeneous 3-manifolds

We solve the Bonnet problem for surfaces in the homogeneous 3-manifolds with a 4-dimensional isometry group. More specifically, we show that a simply connected real analytic surface in H^2xR or S^2xR is uniquely determined pointwise by its metric and its principal curvatures if and only if it is not a minimal or a properly helicoidal surface. In the remaining three types of homogeneous 3-manifolds, we show that except for constant mean curvature surfaces and helicoidal surfaces, all simply connected real analytic surfaces are pointwise determined by their metric and principal curvatures.

math.DG↗
arXiv2006

Harmonic maps and constant mean curvature surfaces in $\H^2 \times \R$

We introduce a hyperbolic Gauss map into the Poincare disk for any surface in H^2xR with regular vertical projection, and prove that if the surface has constant mean curvature H=1/2, this hyperbolic Gauss map is harmonic. Conversely, we show that every nowhere holomorphic harmonic map from an open simply connected Riemann surface into the Poincare disk is the hyperbolic Gauss map of a two-parameter family of such surfaces. As an application we obtain that any holomorphic quadratic differential on the surface can be realized as the Abresch-Rosenberg holomorphic differential of some, and generically infinitely many, complete surfaces with H=1/2 in H^2xR. A similar result applies to minimal surfaces in the Heisenberg group Nil_3. Finally, we classify all complete minimal vertical graphs in H^2xR.

math.DG↗
arXiv2006

A characterization of constant mean curvature surfaces in homogeneous 3-manifolds

It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manifolds isometric to H^2xR or having isometry group isomorphic either to the one of the universal cover of PSL(2,R), or to the one of a certain class of Berger spheres. It turns out that, except for the case of these Berger spheres, there exist some exceptional surfaces with holomorphic Hopf differential and non-constant mean curvature.

math.DG↗