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Jose M. Espinar

Publications and source records attributed to Jose M. Espinar.

16 recordsLinked to original sources
arXiv2017

On the structure of complete 3-manifolds with nonnegative scalar curvature

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 $Σ\subset \mathcal{N} $ is a complete orientable connected area-minimizing cylinder so that $π_1 (Σ) \in π_1 (\mathcal{N})$. 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 $π_1 (\mathcal{N})$ 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).

math.DG↗
arXiv2015

Gradient Schrödinger Operators, Manifolds with Density and applications

The aim of this paper is twofold. On the one hand, the study of gradient Schrödinger operators on manifolds with density $ϕ$. We classify the space of solutions when the underlying manifold is $ϕ-$parabolic. As an application, we extend the Naber-Yau Liouville Theorem, and we will prove that a complete manifold with density is $ϕ-$parabolic if, and only if, it has finite $ϕ-$capacity. Moreover, we show that the linear space given by the kernel of a nonnegative gradient Schrödinger operators is one dimensional provided there exists a bounded function on it and the underlying manifold is $ϕ-$parabolic. On the other hand, the topological and geometric classification of complete weighted $H_ϕ-$stable hypersurfaces immersed in a manifold with density $(\amb , g, ϕ)$ satisfying a lower bound on its Bakry-Émery-Ricci tensor. Also, we classify weighted stable surfaces in a three-manifold with density whose Perelman scalar curvature, in short, P-scalar curvature, satisfies $\scad + \frac{\abs{\nabla ϕ}^2 }{4} \geq 0$. Here, the P-scalar curvature is defined as $\scad = R - 2 Δ_g ϕ- \abs{\nabla _g ϕ}^2$, 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.

math.DG↗
arXiv2014

Halfspace type Theorems for Self-Shrinkers

In this short paper we extend the classical Hoffman-Meeks Halfspace Theorem to self-shrinkers, that is: "Let $P $ be a hyperplane passing through the origin. The only properly immersed self-shrinker $Σ$ contained in one of the closed half-space determined by $P$ is $Σ= P$." Our proof is geometric and uses a catenoid type hypersurface discovered by Kleene-Moller. Also, using a similar geometric idea, we obtain that the only complete self-shrinker properly immersed in an closed cylinder $ \overline{B ^{k+1} (R)} \times \mathbb{R}^{n-k}\subset \mathbb R^{n+1}$, for some $k\in \{1, \ldots ,n\}$ and radius $R$, $R \leq \sqrt{2k}$, is the cylinder $\mathbb S ^k (\sqrt{2k}) \times \mathbb{R}^{n-k}$. We also extend the above results for $λ-$hypersurfaces.

math.DG↗
arXiv2011

The space of Constant Mean Curvature surfaces in compact Riemannian Manifolds

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.

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arXiv2011

Finite index operators on surfaces

We consider differential operators $L$ acting on functions on a Riemannian surface, $Σ$, of the form $$L = Δ+ V -a K ,$$where $Δ$ is the Laplacian of $Σ$, $K$ is the Gaussian curvature, $a$ is a positive constant and $V \in C^{\infty}(Σ)$. Such operators $L$ arise as the stability operator of $Σ$ 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 $Σ$. If the potential, $V:= c + P $ with $c$ a nonnegative constant, verifies either an integrability condition, i.e. $P \in L^1(Σ)$ and $P$ is non positive, or a decay condition with respect to a point $p_0 \in Σ$, i.e. $|P(q)|\leq M/d(p_0,q)$ (where $d$ is the distance function in $Σ$), we control the topology and conformal type of $Σ$. Moreover, we establish a {\it Distance Lemma}. We apply such results to complete oriented stable $H-$surfaces immersed in a Killing submersion.

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arXiv2011

Rigidity of stable cylinders in three-manifolds

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.

math.DG↗
arXiv2010

When strictly locally convex hypersurfaces are embedded

In this paper we will prove Hadamard-Stoker type theorems in the following ambient spaces: $\man ^n \times \r$, where $\man ^n $ is a $1/4-$pinched manifold, and certain Killing submersions, e.g., Berger spheres and Heisenberg spaces. That is, under the condition that the principal curvatures of an immersed hypersurfaces are greater than some non-negative constant (depending on the ambient space), we prove that such a hypersurface is embedded and we also study its topology.

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arXiv2010

Locally convex surfaces immersed in a Killing submersion

We generalize Hadamard-Stoker-Currier Theorems for surfaces immersed in a Killing submersion over a strictly Hadamard surface whose fibers are the trajectories of a unit Killing field. We prove that every complete surface whose principal curvatures are greater than a certain function (depending on the ambient manifold) at each point, must be properly embedded, homeomorphic to the sphere or to the plane and, in the latter case, we study the behavior of the end.

math.DG↗
arXiv2009

A Colding-Minicozzi Stability inequality and its applications

We consider operators $L$ acting on functions on a Riemannian surface, $Σ$, of the form $L = Δ+ V +a K.$ Here $Δ$ is the Laplacian of $Σ$, $V$ a non-negative potential on $Σ$, K the Gaussian curvature and $a$ is a non-negative constant. Such operators $L$ arise as the stability operator of $Σ$ 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 $Σ$ 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 $Σ$ and a distance (to the boundary) lemma.

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arXiv2009

Fatou's Theorem and minimal graphs

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.

math.DG↗
arXiv2008

Invariant conformal metrics on S^n

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 $σ_k -$Yamabe type problem for obtaining existence of rotational spheres and Delaunay-type hypersurfaces for some classes of Weingarten hypersurfaces in $\h ^{n+1}$.

math.DG↗
arXiv2008

Complete Constant Mean Curvature surfaces and Bernstein type Theorems in $\mathbb{M}^2\times \mathbb{R}$

In this paper we study constant mean curvature surfaces $Σ$ in a product space, $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function $ν= \meta{N}{\partial_t}$ does not change sign on $Σ$. We classify these surfaces according to the infimum $c(Σ)$ of the Gaussian curvature of the projection of $Σ$. When $H \neq 0$ and $c(Σ)\geq 0$, then $Σ$ is a cylinder over a complete curve with curvature 2H. If H=0 and $c(Σ) \geq 0$, then $Σ$ must be a vertical plane or $Σ$ is a slice $\mathbb{M}^2 \times {t}$, or $\mathbb{M}^2 \equiv \mathbb{R}^2$ with the flat metric and $Σ$ is a tilted plane (after possibly passing to a covering space). When $c(Σ)<0$ and $H>\sqrt{-c(Σ)} /2$, then $Σ$ is a vertical cylinder over a complete curve of $\mathbb{M}^2$ of constant geodesic curvature $2H$. This result is optimal. We also prove a non-existence result concerning complete multi-graphs in $\mathbb{M}^2\times \mathbb{R}$, when $c(\mathbb{M}^2)<0$.

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arXiv2007

Complete surfaces with positive extrinsic curvature in product spaces

We prove that every complete connected immersed surface with positive extrinsic curvature $K$ in $H^2\times R$ must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study the behavior of the end. Then, we focus our attention on surfaces with positive constant extrinsic curvature ($K-$surfaces). We establish that the only complete $K-$surfaces in $S^2\times R$ and $H^2\times R$ are rotational spheres. Here are the key steps to achieve this. First height estimates for compact $K-$surfaces in a general ambient space $M^2\times R$ with boundary in a slice are obtained. Then distance estimates for compact $K-$surfaces (and H-$surfaces) in $H^2\times R$ with boundary on a vertical plane are obtained. Finally we construct a quadratic form with isolated zeroes of negative index.

math.DG↗
arXiv2005

Complete surfaces of constant curvature in H2xR and S2xR

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces H2xR and S2xR. In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c>0 into H2xR and a unique one into S2xR when c>1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c<-1 cannot be isometrically immersed into H2xR or S2xR.

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