arXiv subjects

Jaime Bruck Ripoll

Publications and source records attributed to Jaime Bruck Ripoll.

4 recordsLinked to original sources

arXiv2026

Invariant solutions for the asymptotic Plateau problem in $\mathbb{H}^3$

In this paper, we present solutions to the asymptotic Plateau problem in the hyperbolic space $\mathbb{H}^3$. In this context, we exhibit solutions for curves that are invariant under the action of a one-parameter subgroup of isometries of $\mathbb{H}^3$. To achieve this, we prove the existence of foliations of $\mathbb{H}^3$ by minimal surfaces that are properly embedded, complete, and invariant under these subgroups, which are then used to solve the problem.

math.DG↗

arXiv2018

On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations

We study the Dirichlet problem for the following prescribed mean curvature PDE $$ \begin{cases} -\operatorname{div}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v) \text{ in }Ω\\ v=φ\text{ on }\partialΩ. \end{cases} $$ where $Ω$ is a domain contained in a complete Riemannian manifold $M,$ $f:Ω\times\mathbb{R\rightarrow R}$ is a fixed function and $φ$ is a given continuous function on $\partialΩ$. This is done in three parts. In the first one we consider this problem in the most general form, proving the existence of solutions when $Ω$ is a bounded $C^{2,α}$ domain, under suitable conditions on $f$, with no restrictions on $M$ besides completeness. In the second part we study the asymptotic Dirichlet problem when $M$ is the hyperbolic space $\mathbb{H}^n$ and $Ω$ is the whole space. This part uses in an essential way the geometric structure of $\mathbb{H}^n$ to construct special barriers which resemble the Scherk type solutions of the minimal surface PDE. In the third part one uses these Scherk type graphs to prove the non existence of isolated asymptotic boundary singularities for global solutions of this Dirichlet problem.

math.DG↗

arXiv2016

On the nature of isolated asymptotic singularities of solutions of a family of quasi-linear elliptic PDE's on a Cartan-Hadamard manifold

Let $M$ be a Cartan-Hadamard manifold with sectional curvature satisfying $-b^2\leq K\leq -a^2<0$, $b\geq a>0.$ Denote by $\partial_{\infty}M$ the asymptotic boundary of $M$ and by $\bar M:= M\cup\partial_\infty M$ the geometric compactification of $M$ with the cone topology. We investigate here the following question: Given a finite number of points $p_{1},...,p_{k}\in\partial_\infty M,$ if $u\in C^{\infty}(M)\cap C^{0}\left( \bar{M}\backslash\left\{ p_{1},...,p_{k}\right\} \right) $ satisfies a PDE $\mathcal Q(u)=0$ in $M$ and if $u|_{\partial_\infty M\setminus\left\{p_{1},...,p_{k}\right\}}$ extends continuously to $p_{i},$ $i=1,...,k,$ can one conclude that $u\in C^{0}\left( \bar{M}\right )?$ When $\dim M=2$, for $\mathcal Q$ belonging to a linearly convex space of quasi-linear elliptic operators $\mathcal{S}$ of the form $$ \mathcal{Q}(u)=\operatorname{div}\left( \frac{\mathcal{A}(|\nabla u|)}{|\nabla u|} \nabla u \right)=0, $$ where $\mathcal{A}$ satisfies some structural conditions, then the answer is yes provided that $\mathcal{A}$ has a certain asymptotic growth. This condition includes, besides the minimal graph PDE, a class of minimal type PDEs. In the hyperbolic space $ \mathbb{H}^n$, $n\ge 2,$ we are able to give a complete answer: we prove that $\mathcal{S}$ splits into two disjoint classes of minimal type and $p-$Laplacian type PDEs, $p>1,$ where the answer is yes and no respectively. These two classes are determined by the asymptotic behaviour of $\mathcal A.$ Regarding the class where the answer is negative, we obtain explicit solutions having an isolated non removable singularity at infinity.

math.DG↗

arXiv2014

An extension of Ruh-Vilms Theorem for hypersurfaces in symmetric spaces and some applications

The main purpose of the paper is twofold: First, to extend a well known theorem of Ruh-Vilms in the Euclidean space to symmetric spaces and, secondly, to apply this result to extend Hoffman-Osserman-Schoen Theorem (HOS Theorem) to 3-dimensional symmetric spaces. Precisely, it is defined a Gauss map of a hypersurface M^{n-1} immersed in a symmetric space N^n taking values in the unit pseudo sphere S^m of the Lie algebra g of the isometry group of N, dim(g)=m+1, and it is proved that M has CMC if and only if its Gauss map is harmonic. As an application, it is proved that if dim(N)=3 and the image of the Gauss map of a CMC surface S immersed in N is contained in a hemisphere of S^m determined by a vector X, then S is invariant by the one parameter subgroup of isometries of N of the Killing field determined by X. In particular, it is obtained an extension of HOS Theorem to the 3-dimensional hyperbolic space. In the paper it is also shown that the holomorphic quadratic form induced by the Gauss map coincides (up to a sign) with the Hopf quadratic form when the ambient space is a space form and with the Abresch-Rosenberg quadratic form when the ambient space is H^2 x R and S^2 x R providing, then, an unified way of relating Hopf's and Abresch-Rosenberg's quadratic form with the quadratic form induced by a harmonic Gauss map of a CMC surface in these 5 spaces.

math.DG↗