Gradient Schrodinger operators, manifolds with density and applications

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:455

Article

Espinar, Jose M.¹ 

[1] Inst Matematica Pura & Aplicada, Rio De Janeiro, Brazil

关键词: Manifolds with density;    Generalized mean curvature;    Stability;    Schrodinger operators;    Geometric flows;    Optimal transport;   

DOI  :  10.1016/j.jmaa.2017.06.055

来源: Elsevier

【 摘 要 】

The aim of this paper is twofold. On the one hand, the study of gradient Schrodinger 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 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 Schriidinger operator is one dimensional provided there exists a bounded function in the kernel and the underlying manifold is phi-parabolic. On the other hand, the topological and geometrical classification of complete weighted Ho-stable hypersurfaces immersed in a manifold with density (N, g, phi) satisfying a lower bound on its Bakry-Emery-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 V-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 phi-parabolic manifolds. In particular, we recover the case of the Gaussian measure. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free