Gradient Schrödinger operators, manifolds with density and applications

Abstract

The aim of this paper is twofold. On the one hand, the study of gradient Schrödinger operators on manifolds with density ϕ. We classify the space of solutions when the underlying manifold is ϕ-parabolic. As an application, we extend the Naber–Yau Liouville Theorem and we prove that a complete manifold with density is ϕ-parabolic if, and only if, it has finite ϕ-capacity. Moreover, we show that the linear space given by the kernel of a nonnegative gradient Schrödinger operator is one dimensional provided there exists a bounded function in the kernel and the underlying manifold is ϕ-parabolic. On the other hand, the topological and geometrical classification of complete weighted Hϕ-stable hypersurfaces immersed in a manifold with density (N,g,ϕ) satisfying a lower bound on its Bakry–Émery–Ricci tensor. Also, we classify weighted stable surfaces in a three-manifold with density whose Perelman scalar curvature, in short, P-scalar curvature, together with the squared norm of the gradient of the density satisfy a lower bound. Finally, we discuss the relationship of manifolds with density, Mean Curvature Flow (MCF), Ricci Flow and Optimal Transportation Theory. We obtain classification results for stable self-similar solutions to the MCF, and also for stable translating solitons to the MCF (as far as we know, the first classification result). Moreover, for gradient Ricci solitons, we recover the Hamilton–Ivey–Perelman classification assuming only a L2-type bound on the scalar curvature and a inequality between the scalar curvature and the Ricci tensor. We also classify gradient Ricci solitons when the scalar curvature does not change sign. We finish by classifying critical transportation plans for the Boltzman entropy on ϕ-parabolic manifolds. In particular, we recover the case of the Gaussian measure.