梯度薛定谔算子、密度流形及其应用
Gradient Schrödinger Operators, Manifolds with Density and applications
摘要：
该文致力于研究两个问题.一方面，研究了密度为$ϕ$的流形上的梯度薛定谔算子。当基本流形是$ϕ-$抛物流形时，我们对解空间进行了分类.作为应用，我们推广了Naber-Yau Liouville定理，证明了一个具有密度的完备流形是$ϕ-$抛物流形的充要条件是它具有有限的$ϕ-$容量.此外，我们还证明了由非负梯度薛定谔算子的核给出的线性空间是一维的，只要在其上存在一个有界函数，并且其下的流形是$ϕ-$抛物流形.另一方面，研究了密度为$(\amb , g, ϕ)$的流形中的完备加权$H_ϕ-$稳定超曲面的拓扑和几何分类，其中$(\amb , g, ϕ)$满足其Bakry-Émery-Ricci张量的下界。同时，我们还对密度为Perelman标量曲率，简称P-数量曲率满足$\scad + \frac{\abs{\nabla ϕ}^2 }{4} \geq 0$的三流形中的加权稳定曲面进行了分类.这里，P-数量曲率被定义为$\scad = R - 2 Δ_g ϕ- \abs{\nabla _g ϕ}^2$，即$R$是$(\amb ,g)$的数量曲率.最后讨论了流形与密度、平均曲率流、Ricci流和最优运输理论的关系。特别地，我们得到了MCF的稳定自相似解的分类结果，也得到了MCF的稳定平移孤子的分类结果，据我们所知，这是关于稳定平移孤子的第一个分类结果.
英文摘要：
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 will 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 operators is one dimensional provided there exists a bounded function on it and the underlying manifold is $ϕ-$parabolic. On the other hand, the topological and geometric classification of complete weighted $H_ϕ-$stable hypersurfaces immersed in a manifold with density $(\amb , g, ϕ)$ 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, satisfies $\scad + \frac{\abs{\nabla ϕ}^2 }{4} \geq 0$. Here, the P-scalar curvature is defined as $\scad = R - 2 Δ_g ϕ- \abs{\nabla _g ϕ}^2$, 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.
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