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.

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.

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.

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.

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.

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.

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 $a leq 1/4$

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 $H^2xR$.