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Search Results: 1 - 10 of 8270 matches for " Jose Espinar "
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Gradient Schr?dinger Operators, Manifolds with Density and applications
Jose M. Espinar
Mathematics , 2012,
Abstract: The aim of this paper is twofold. On the one hand, the study of gradient Schr\"{o}dinger operators on manifolds with density $\phi$. We classify the space of solutions when the underlying manifold is $\phi-$parabolic. As an application, we extend the Naber-Yau Liouville Theorem, and we will prove that a complete manifold with density is $\phi -$parabolic if, and only if, it has finite $\phi-$capacity. Moreover, we show that the linear space given by the kernel of a nonnegative gradient Schr\"{o}dinger operators is one dimensional provided there exists a bounded function on it and the underlying manifold is $\phi -$parabolic. On the other hand, the topological and geometric classification of complete weighted $H_\phi -$stable hypersurfaces immersed in a manifold with density $(\amb , g, \phi)$ satisfying a lower bound on its Bakry-\'{E}mery-Ricci tensor. Also, we classify weighted stable surfaces in a three-manifold with density whose Perelman scalar curvature, in short, P-scalar curvature, satisfies $\scad + \frac{\abs{\nabla \phi}^2 }{4} \geq 0$. Here, the P-scalar curvature is defined as $\scad = R - 2 \Delta _g \phi - \abs{\nabla _g \phi }^2$, being $R$ the scalar curvature of $(\amb ,g)$. Finally, we discuss the relationship of manifolds with density, Mean Curvature Flow (MCF), Ricci Flow and Optimal Transportation Theory. In particular, we obtain classification results for stable self-similiar solutions to the MCF, and also for stable translating solitons to the MCF, as far as we know, this is the first classification result on stable translating solitons.
Invariant conformal metrics on S^n
Jose M. Espinar
Mathematics , 2008,
Abstract: In this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on the sphere which are invariant by a $k-$parameter subgroup of conformal diffeomorphisms of the sphere, giving a bound on its maximum dimension. Moreover, we classify conformal metrics on the sphere whose eigenvalues of the Shouten tensor are all constant (we call them \emph{isoparametric conformal metrics}), and we use a classification result for radial conformal metrics which are solution of some $\sigma _k -$Yamabe type problem for obtaining existence of rotational spheres and Delaunay-type hypersurfaces for some classes of Weingarten hypersurfaces in $\h ^{n+1}$.
Finite index operators on surfaces
Jose M. Espinar
Mathematics , 2009,
Abstract: We consider differential operators $L$ acting on functions on a Riemannian surface, $\Sigma$, of the form $$L = \Delta + V -a K ,$$where $\Delta$ is the Laplacian of $\Sigma$, $K$ is the Gaussian curvature, $a$ is a positive constant and $V \in C^{\infty}(\Sigma)$. Such operators $L$ arise as the stability operator of $\Sigma$ immersed in a Riemannian three-manifold with constant mean curvature (for particular choices of $V$ and $a$). We assume $L$ is nonpositive acting on functions compactly supported on $\Sigma$. If the potential, $V:= c + P $ with $c$ a nonnegative constant, verifies either an integrability condition, i.e. $P \in L^1(\Sigma)$ and $P$ is non positive, or a decay condition with respect to a point $p_0 \in \Sigma$, i.e. $|P(q)|\leq M/d(p_0,q)$ (where $d$ is the distance function in $\Sigma$), we control the topology and conformal type of $\Sigma$. Moreover, we establish a {\it Distance Lemma}. We apply such results to complete oriented stable $H-$surfaces immersed in a Killing submersion.
Rigidity of stable cylinders in three-manifolds
Jose M. Espinar
Mathematics , 2011,
Abstract: In this paper we show how the existence of a certain stable cylinder determines (locally) the ambient manifold where it is immersed. This cylinder has to verify a {\it bifurcation phenomena}, we make this explicit in the introduction. In particular, the existence of such a stable cylinder implies that the ambient manifold has infinite volume.
On the structure of complete 3-manifolds with nonnegative scalar curvature
Jose M. Espinar
Mathematics , 2011,
Abstract: In this paper we will show the following result: Let $\mathcal{N} $ be a complete (noncompact) connected orientable Riemannian three-manifold with nonnegative scalar curvature $S \geq 0$ and bounded sectional curvature $ K_{s} \leq K $. Suposse that $\Sigma \subset \mathcal{N} $ is a complete orientable connected area-minimizing cylinder so that $\pi_1 (\Sigma) \in \pi_1 (\mathcal{N})$. Then $\mathcal{N}$ is locally isometric either to $\mathbb{S} ^1 \times \mathbb{R} ^2 $ or $\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{R}$ (with the standard product metric). As a corollary, we will obtain: Let $\mathcal{N} $ be a complete (noncompact) connected orientable Riemannian three-manifold with nonnegative scalar curvature $S \geq 0$ and bounded sectional curvature $ K_{s} \leq K $. Assume that $\pi_1 (\mathcal{N})$ contains a subgroup which is isomorphic to the fundamental group of a compact surface of positive genus. Then, $\mathcal{N}$ is locally isometric to $\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{R}$ (with the standard product metric).
The space of Constant Mean Curvature surfaces in compact Riemannian Manifolds
Jose M. Espinar
Mathematics , 2011,
Abstract: The main point of this paper is that, under suitable conditions on the mean curvature and the Ricci curvature of the ambient space, we can extend Choi-Schoen's Compactness Theorem to compact embedded minimal surfaces to simple immersed compact H-surfaces in a Riemannian manifold with positive Ricci curvature (the mean curvature small depending on the Ricci curvature). Also, we prove that the space of convex embedded (fixed) constant mean curvature hypersurfaces in a simply connected 1/4-pinched manifold is compact.
Totally umbilical disks and applications to surfaces in three-dimensional homogeneous spaces
Jose M. Espinar,Isabel Fernandez
Mathematics , 2009,
Abstract: Following ideas of Choe and Fernandez-do Carmo, we give sufficient conditions for a disk type surface, with piecewise smooth boundary, to be totally umbilical for a given Coddazi pair. As a consequence, we obtain rigidity results for surfaces in space forms and in homogeneous product spaces that generalizes some known results.
Complete Constant Mean Curvature surfaces in homogeneous spaces
Jose M. Espinar,Harold Rosenberg
Mathematics , 2009,
Abstract: In this paper we classify complete surfaces of constant mean curvature whose Gaussian curvature does not change sign in a simply connected homogeneous manifold with a 4-dimensional isometry group.
Fatou's Theorem and minimal graphs
Jose M. Espinar,Harold Rosenberg
Mathematics , 2009,
Abstract: In this paper we extend a recent result of Collin-Rosenberg ({\it a solution to the minimal surface equation in the Euclidean disc has radial limits almost everywhere}) to a large class of differential operators in Divergence form. Moreover, we construct an example (in the spirit of \cite{CR2}) of a minimal graph in $\mr$, where $\m$ is a Hadamard surface, over a geodesic disc which has finite radial limits in a mesure zero set.
A Colding-Minicozzi Stability inequality and its applications
Jose M. Espinar,Harold Rosenberg
Mathematics , 2008,
Abstract: We consider operators $L$ acting on functions on a Riemannian surface, $\Sigma$, of the form $L = \Delta + V +a K.$ Here $\Delta$ is the Laplacian of $\Sigma$, $V$ a non-negative potential on $\Sigma$, K the Gaussian curvature and $a$ is a non-negative constant. Such operators $L$ arise as the stability operator of $\Sigma$ immersed in a Riemannian 3-manifold with constant mean curvature (for particular choices of $V$ and $a$). We assume L is nonpositive acting on functions compactly supported on $\Sigma$ and we obtain results in the spirit of some theorems of Ficher-Colbrie-Schoen, Colding-Minicozzi, and Castillon. We extend these theorems to $a \leq 1/4$. We obtain results on the conformal type of $\Sigma$ and a distance (to the boundary) lemma.
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