Jaime Ripoll

Publications (8)1.47 Total impact

• Article: Regularity at infinity of Hadamard manifolds with respect to some elliptic operators and applications to asymptotic Dirichlet problems

  Jaime Ripoll, Miriam Telichevesky
  [show abstract] [hide abstract]
  ABSTRACT: Let $M$ be Hadamard manifold with sectional curvature $K_{M}\leq-k^{2}$, $k>0$. Denote by $\partial_{\infty}M$ the asymptotic boundary of $M$. We say that $M$ satisfies the strict convexity condition (SC condition) if, given $x\in\partial_{\infty}M$ and a relatively open subset $W\subset\partial_{\infty}M$ containing $x$, there exists a $C^{2}$ open subset $\Omega\subset M$ such that $x\in\operatorname*{Int}(\partial_{\infty}\Omega) \subset W$ and $M\setminus\Omega$ is convex. We prove that the SC condition implies that $M$ is regular at infinity relative to the operator $$\mathcal{Q}[u] :=\mathrm{{div}}(\frac{a(|\nabla u|)}{|\nabla u|}\nabla u),$$ subject to some conditions. It follows that under the SC condition, the Dirichlet problem for the minimal hypersurface and the $p$-Laplacian ($p>1$) equations are solvable for any prescribed continuous asymptotic boundary data. It is also proved that if $M$ is rotationally symmetric or if $\inf_{B_{R+1}}K_{M}\geq-e^{2kR}/R^{2+2\epsilon}, R\geq R^{\ast},$ for some $R^{\ast}$ and $\epsilon>0,$ where $B_{R+1}$ is the geodesic ball with radius $R+1$ centered at a fixed point of $M,$ then $M$ satisfies the SC condition.
  01/2013;
• Article: An interior gradient estimate for the mean curvature equation of Killing graphs

  Marcos Dajczer, Jorge H. Lira, Jaime Ripoll
  [show abstract] [hide abstract]
  ABSTRACT: We extend the interior gradient estimate due to Korevaar-Simon for solutions of the mean curvature equation from the case of Euclidean hypersurfaces to the general case of Killing graphs. As an application, we prove the existence and uniqueness of radial graphs in hyperbolic space with prescribed mean curvature function and asymptotic boundary data at infinity.
  06/2012;
• Article: An extension of a theorem of Serrin to graphs in warped products

  Marcos Dajczer, Jaime Ripoll
  [show abstract] [hide abstract]
  ABSTRACT: In this article we extend a well known theorem of J. Serrin about existence and uniqueness of graphs of constant mean curvature in Euclidean space to a broad class of Riemannian manifolds. Our result also generalizes several others proved recently and includes the new case of Euclidean “rotational” graphs with constant mean curvature.
  Journal of Geometric Analysis 04/2012; 15(2):193-205. · 0.76 Impact Factor
• Source

  Available from: preprint.impa.br

  Article: A Bernstein-type theorem for Riemannian manifolds with a Killing field

  Luis J. Alías, Marcos Dajczer, Jaime Ripoll
  [show abstract] [hide abstract]
  ABSTRACT: The classical Bernstein theorem asserts that any complete minimal surface in Euclidean space \mathbbR3\mathbb{R}^3 that can be written as the graph of a function on \mathbbR2\mathbb{R}^2 must be a plane. In this paper, we extend Bernstein’s result to complete minimal surfaces in (may be non-complete) ambient spaces of non-negative Ricci curvature carrying a Killing field. This is done under the assumption that the sign of the angle function between a global Gauss map and the Killing field remains unchanged along the surface. In fact, our main result only requires the presence of a homothetic Killing field.
  Annals of Global Analysis and Geometry 04/2012; 31(4):363-373. · 0.71 Impact Factor
• Source

  Available from: ArXiv

  Article: Asymptotic Dirichlet problems for Laplace's and minimal equations on Hadamard manifolds

  Jaime Ripoll, Miriam Telichevesky
  [show abstract] [hide abstract]
  ABSTRACT: It is proved the existence of entire solutions of the Laplace's and minimal hypersurface's PDEs on a Hadamard manifold $M$ under certain curvature conditions by investigating the asymptotic Dirichlet's problems for these PDEs. In the harmonic case it is obtained an existence result which assumes the same growth condition on the sectional curvature as of Theorem 1.2 of E. Hsu \cite{Hsu} but that contemplates cases having Ricci curvature with exponential decay. It is also obtained a result which extends and improves Theorem 3.6 of Choi \cite{Choi}. In the minimal case one obtains an extension and an improvement of Theorem 1 of N. do Esp\'{\i}rito-Santo, S. Fornari and J. Ripoll \cite{EFR}, and partial extensions of Theorem 5.2 of J. A. G\'alvez and H. Rosenberg \cite{GR} by allowing the sectional curvature of $M$ degenerate to 0 at infinity.
  01/2011;
• Source

  Available from: ArXiv

  Article: Constant mean curvature hypersurfaces with single valued projections on planar domains

  Marcos Dajczer, Jaime Ripoll
  [show abstract] [hide abstract]
  ABSTRACT: A classical problem in constant mean curvature hypersurface theory is, for given $H\geq 0$, to determine whether a compact submanifold $\Gamma^{n-1}$ of codimension two in Euclidean space $\R_+^{n+1}$, having a single valued orthogonal projection on $\R^n$, is the boundary of a graph with constant mean curvature $H$ over a domain in $\R^n$. A well known result of Serrin gives a sufficient condition, namely, $\Gamma$ is contained in a right cylinder $C$ orthogonal to $\R^n$ with inner mean curvature $H_C\geq H$. In this paper, we prove existence and uniqueness if the orthogonal projection $L^{n-1}$ of $\Gamma$ on $\R^n$ has mean curvature $H_L\geq-H$ and $\Gamma$ is contained in a cone $K$ with basis in $\R^n$ enclosing a domain in $\R^n$ containing $L$ such that the mean curvature of $K$ satisfies $H_K\geq H$. Our condition reduces to Serrin's when the vertex of the cone is infinite.
  05/2010;
• Source

  Available from: ArXiv

  Article: A topological pinching for the injectivity radius of a compact surface in S^3 and in H^3

  Edson S. Figueiredo, Jaime Ripoll
  [show abstract] [hide abstract]
  ABSTRACT: It is given a topological pinching for the injectivity radius of a compact embedded surface either in the sphere or in the hyperbolic space
  08/2008;
• Source

  Available from: psu.edu

  Article: Complete Conical Type End Immersed Manifolds in R^N

  Jaime Ripoll
  [show abstract] [hide abstract]
  ABSTRACT: this paper we introduce and describe the topology of a family of immersed manifolds in R having a nice behaviour at infinity, which we call conical type end manifolds, defined as follows. Let M be a complete non compact nGammadimensional Riemannian manifold, and let OE : M ! R be an isometric immersion. As usual, we identify M with OE(M) and assume that 0 = 2 M: Given p 2 M we denote by N(p) the orthogonal projection of p=jpj over T p M ; where T p M is the tangent space of M at p and T p M its orthogonal complement in R : Given ff 0; we say that M is a ffGammaconical type end immersed manifold of R if ffi ff := lim d(p;p0 ) !1 (p; p 0 )jN(p)j ! 1 (1) where d(p; p 0 ) is the intrinsic distance in M from p to an arbitrary but fixed point p 0 in M: An obvious example is as follows. Setting S (1) = fx 2 R j jxj = 1g; take an immersed compact manifold V nGamma1 ae S (1); and let M be the part of the cone over V exterior to S (1); that is, M = ftx j t 1; x 2 V g: Since N(p) = 0 for all p 2 M; it follows that M is an immersed ff-conical type end manifold of R for any ff 0 once one "completes" M 183 with some compact immersed nGamma manifold of R : Roughly speaking, this example is typical at least in the case ff ? 0 in the sense that a ffGammaconical type end immersed manifold M; ff ? 0; is uniformly asymptotic to a cone over a compact subset of the sphere (Theorem 1.1 (d))
  09/1998;