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.

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 $\h ^{n+1}$.

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.

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.

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).

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.

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.

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.

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.

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.