Gradient Schr\"odinger Operators, Manifolds with Density and applications

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.