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# On a fully nonlinear version of the Min-Oo Conjecture

## José María Espinar  Universidad de Cádiz

In this talk, we prove that the Min-Oo's conjecture holds if we consider a compact connected locally conformally flat manifold with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality, and the mean curvature of the boundary is controled bellow by the mean curvature of a geodesic ball in the standard unit-sphere. This is a joint work with E. Barbosa and M.P. Cavalcante.

# Superficies Mínimas con Borde

## José María Espinar  Universidad de Cádiz

Introducción y conceptos básicos. Primera fórmula de variación y aplicaciones. Teorema de Nitsche. Aplicaciones conformes. Teorema de Fraser-Schoen. Segunda fórmula de variación y aplicaciones. Teorema de Shiffman.

# Superficies Mínimas con Borde

## José María Espinar  Universidad de Cádiz

Introducción y conceptos básicos. Primera fórmula de variación y aplicaciones. Teorema de Nitsche. Aplicaciones conformes. Teorema de Fraser-Schoen. Segunda fórmula de variación y aplicaciones. Teorema de Shiffman.

# Extremal domains in Hadamard manifolds

## José María Espinar  Universidad de Cádiz

In this talk we investigate the geometry and topology of f-extremal domains in a manifold with negative sectional curvature. A f-extremal domain is a domain that supports a positive solution to the overdetermined elliptic problem \begin{eqnarray} \label{1.3} \left\{ \begin{array}{llll} \Delta{u}+f(u)=0 \quad&\mathrm{in}\quad ~~\Omega,\\ u>0 \quad&\mathrm{in}\quad ~~\Omega,\\ u=0 \quad&\mathrm{on}\quad \partial\Omega,\\ \langle\nabla{u},\vec{v}\rangle_{M}=\alpha \quad&\mathrm{on}\quad \partial\Omega, \end{array} \right. \end{eqnarray}where $\Omega$ is an open connected domain in a complete Hadamard $n$-manifold $(M,g)$ with boundary $\partial\Omega$ of class $C^{2}$, $f$ is a given Lipschitz function, $\langle\cdot,\cdot\rangle_{M}$ is the inner product on $M$ induced by the metric $g$, and $\alpha$, $\vec{v}$ the unit outward normal vector of the boundary $\partial\Omega$ and $\alpha$ a non-positive constant. We will show narrow properties of such domains in a Hadamard manifolds and characterize the boundary at infinity. We give an upper bound for the Hausdorff dimension of its boundary at infinity. Later, we focus on $f$-extremal domains in the Hyperbolic Space $\mathbb H^n$. Symmetry and boundedness properties will be shown. Hence, we are able to prove the Berestycki-Caffarelli-Nirenberg Conjecture in $\mathbb H^2$. Specifically: Let $\Omega \subset \mathbb H^2$ a domain with properly embedded $C^2$ connected boundary such that $\mathbb H^2 \setminus \overline{\Omega}$ is connected. If there exists a (strictly) positive function $u\in{C}^{2}(\Omega)$ that solves the equation \begin{eqnarray*} \left\{ \begin{array}{llll} \Delta{u}+f(u)=0 \quad&\mathrm{in}\quad ~~\Omega,\\ u>0 \quad&\mathrm{in}\quad ~~\Omega,\\ u=0 \quad&\mathrm{on}\quad \partial\Omega,\\ \langle\nabla{u},\vec{v}\rangle_{\mathbb H ^2}=\alpha \quad&\mathrm{on}\quad \partial\Omega, \end{array} \right. \end{eqnarray*} where $f:(0,+\infty)\rightarrow\mathbb{R}$ satisfies $f(t)\geq \lambda \, t$ for some constant $\lambda$ satisfying $\lambda> \frac{1}{4}$, then $\Omega$ must be a geodesic ball and $u$ radially symmetric. If time permits, we will generalize the above results to more general OEPs.

# Compactness result for apparent horizont

## José María Espinar  Universidad de Cádiz

In this talk we will extend the Choi-Schoen [Invent. Math. 81 (1985) 387--394] compactness result for minimal surfaces to apparent horizons in the context of General Relativity. We also discuss some applications of the aforementioned result.

# Operadores L = Δ + V - aK con potencial integrable

## José María Espinar  Universidad de Cádiz

En esta charla estudiaremos operadores del tipo L := Δ + V - aK en una superficie Riemanniana Σ, tal que V := c + P, donde c es una constante no-negativa y P es una función no-positiva e integrable en Σ. El principal resultado será demostrar si L es no-positivo cuando actúa sobre funciones no-negativas de soporte compacto, a > 1/4 y Σ es completa, entonces Σ es compacta o parabólica con área finita. Daremos aplicaciones a superficies estables en submersiones.

# On a Colding-Minicozzi Stability-type inequality and its applications

## José María Espinar  Universidad de Cádiz

We consider operator L acting on functions on a Riemannian surface of the form $L = Delta + V - aK$. Here $Delta$ is the Laplacian, $V$ is a non-negative potential, $K$ is the Gaussian curvature and $a$ is a non-negative constant. Such operators $L$ arise as the stability operator of immersed in a Riemannian 3-manifold with constant mean curvature (for particular choices of $V$ and $a$). We assume $L$ is non-positive acting on functions compactly supported and we obtain results in the spirit of some theorems of Ficher-Colbrie-Shoen, Colding-Minicozzi and Castillon. We extend these theorems to $a leq 1/4$

# Teorema de Hadamard-Stoker en $\mathbb{H}^2\times \mathbb{R}$

## José María Espinar  Universidad de Cádiz

El clásico Teorema de Hadamard-Stoker establece que: Toda superficie completa y localmente estrictamente convexa en $R^3$ es embebida. Además es homeomorfa a una esfera si es cerrada o a un plano si es abierta. Luego, el objetivo será extender dicho resultado para superficies completas con curvatura extrínseca positiva en $H^2xR$.

# José María Espinar

Conferencias impartidas
13
Visitas al departamento
17
Última visita
Perfil en Mathscinet Perfil en Zentralblatt País de origen
España

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