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

TL;DR

This paper studies gradient Schr"odinger operators on manifolds with density, classifies solutions and stable hypersurfaces, and explores applications to geometric flows and optimal transportation, providing new classification results.

Contribution

It introduces new classifications of solutions and stable hypersurfaces on manifolds with density, extending Liouville theorems and analyzing geometric flow solitons.

Findings

Classified solutions of gradient Schr"odinger operators on ta-parabolic manifolds.

Extended Naber-Yau Liouville Theorem for manifolds with density.

Provided classification of stable solutions to Mean Curvature Flow and Ricci Flow.

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.