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Luquesio P. Jorge

Publications and source records attributed to Luquesio P. Jorge.

9 recordsLinked to original sources
arXiv2021

Stochastic half-space theorems for minimal surfaces and $H$-surfaces of $\mathbb{R}^{3}$

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface immersed with bounded curvature in $M\times \R_+$ equals some $M\times \{s\}$ provided $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with $\text{Ric}_M \geq 0$ and whose sectional curvatures are bounded from above. For $H$-surfaces we prove that a stochastically complete surface $M$ can not be in the mean convex side of a $H$-surface $N$ embedded in $\R^3$ with bounded curvature if $\sup \vert H_{_M}\vert < H$, or ${\rm dist}(M,N)=0$ when $\sup \vert H_{_M}\vert = H$. Finally, a maximum principle at infinity is shown assuming $M$ has non-empty boundary.

math.DG↗
arXiv2019

The Gauss map of a complete minimal surface with finite total curvature

In [15] Robert Osserman proved that the image of the Gauss map of a complete, non flat minimal surface in R^3 with finite total curvature miss at most 3 points. In this paper we prove that the Gauss map of such a minimal immersions omit at most 2 points. This is a sharp result since the Gauss map of the catenoid omits exactly two points. In fact we prove this result for a wider class of isometric immersions, that share the basic differential topological properties of the complete minimal surfaces of finite total curvature.

math.DG↗
arXiv2019

On the Gauss map of finite geometric type surfaces

Surfaces of finite geometric type are complete, immersed into the tree-dimensional Euclidean space with finite total curvature and Gauss map extending to an oriented compact surface as a smooth branched covering map over the unit sphere of the Euclidean three dimensional space. In a recent preprint J. Jorge and F. Mercuri gave a geometric proof that the Gauss map can not omit three or more points if the immersion is minimal and no flat. Here we give a topological proof of this result in the class of no flat finite geometric type surfaces and also give a topological classification when the Gauss map is a regular covering map. This facts are easy applications of our main result, a generalization of the little Picard theorem for the class of branched covering of a finite geometric type surface into the unit sphere of the tree dimensional Euclidean space. A finite geometric type surface given by a compact surface minus a finite set of points has the following property: any branched covering from the 0surface to the unit Euclidean sphere having a C extension to the compact surfaces can miss at most 2 points. This is a generalization of the little Picard theorem to the class of finite geometric type surfaces.

math.DG↗
arXiv2016

Dirichlet spectrum and Green function

In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/λ_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower estimates for the series $\sum λ_{i}^{-2}(Ω)$ where $Ω$ is an extrinsic ball of a proper minimal surface of $\mathbb{R}^{3}$. In the second part we show that the first eigenvalue of bounded domains is given by iteration of the Green operator and taking the limit, $λ_{1}(Ω)=\lim_{k\to \infty} \Vert G^k(f)\Vert_{2}/\Vert G^{k+1}(f)\Vert_{2}$ for any function $f>0$. In the third part we obtain explicitly the $L^{1}(Ω, μ)$-momentum spectrum of a bounded domain $Ω$ in terms of its Green operator. In particular, we obtain the first eigenvalue of a weighted bounded domain in terms of the $L^{1}(Ω, μ)$-momentum spectrum, extending the work of Hurtado-Markvorsen-Palmer on the first eigenvalue of rotationally invariant balls.

math.DG↗
arXiv2012

On the spectrum of bounded immersions

In this paper, we investigate the relationship between the discreteness of the spectrum of a non-compact, extrinsically bounded submanifold $φ\colon M^m \ra N^n$ and the Hausdorff dimension of its limit set $\limφ$. In particular, we prove that if $φ\colon \!M^2 \ra D \subseteq \R^3$ is a minimal immersion into an open, bounded, strictly convex subset $D$ with $C^2$-boundary, then $M$ has discrete spectrum provided that $\haus_Ψ(\limφ\cap D)=0$, where $\haus_Ψ$ is the generalized Hausdorff measure of order $Ψ(t) = t^2|\log t|$. Our theorem applies to a number of examples recently constructed by various authors in the light of N. Nadirashvili's discovery of complete, bounded minimal disks in $\R^3$, as well as to solutions of Plateau's problems, giving a fairly complete answer to a question posed by S.T. Yau in his Millenium Lectures. Suitable counter-examples show the sharpness of our results: in particular, we develop a simple criterion for the existence of essential spectrum which is suited for the techniques developed after Jorge-Xavier and Nadirashvili's examples.

math.DG↗
arXiv2011

On stochastically complete submanifolds

Using a deep criteria due to Pigola, Rigoli and Setti, we prove that a geodesically complete, properly immersed submanifold M of a stochastically complete Riemannian manifold N is stochastically complete. This implies that the weak Omori-Yau maximum principle holds on M. As geometric application, we prove sectional curvature estimates for properly immersed cilindrically bounded submanifolds.

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arXiv2009

On the essential spectrum of Nadirashvili-Martin-Morales minimal surfaces

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of $\mathbb{R}^{3}$ is discrete. This gives a positive answer to a question of Yau.

math.DG↗
arXiv2002

Properness of minimal surfaces with bounded curvature

We show that immersed minimal surfaces of $\mathbb{R}^{3}$ with bounded curvature and proper self intersections are proper. We also show that the restriction of the immersing map to a wide component is always proper. When the immersing map is injective the whole surface is a wide component. Prior to these results it was only known that injectively immersed minimal surfaces with bounded curvature were proper.

math.DG↗