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Talks by José María Espinar

On a fully nonlinear version of the Min-Oo Conjecture

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.

Seminario 1ª planta, IEMath

Superficies Mínimas con Borde

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.

Seminario 2ª Planta, IEMath-GR

Superficies Mínimas con Borde

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.

Seminario 2ª Planta, IEMath-GR

Extremal domains in Hadamard manifolds

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.

Seminario 1ª Planta, IEMath-Gr

Lectures on stable minimal surfaces II

Universidad de Cádiz

Lectures on stable minimal surfaces I

Universidad de Cádiz

Compactness result for apparent horizont

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.

Seminario de Matemáticas. 1ª Planta, sección de Matemáticas.

Operadores L = Δ + V - aK con potencial integrable

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.

Seminario de Matemáticas. 2ª Planta, sección de Matemáticas.

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

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 $a leq 1/4$

Seminario de Matemáticas. 2ª Planta, sección de Matemáticas.

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

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 $H^2xR$.

Seminario del Dpto. de Análisis Matemático

Representación de Bryant para H=1, parte II

Universidad de Cádiz

Representación de Bryant para H=1

Universidad de Cádiz

José María Espinar

Universidad de Cádiz

Number of talks
13
Number of visits
17
Last visit
Profile in Mathscinet
[Mathscinet]
Profile in Zentralblatt
[zentralblatt]
Country of origin
España

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