ABSTRACT

This paper generalizes to the context of smooth metric measure spaces and submanifolds with negative sectional curvatures some well-known geometric estimates on the p-fundamental tone by using vector fields satisfying a positive divergence condition. Choosing the vector field to be the gradient of an appropriately chosen distance function yields generalised McKean estimates whilst other choices of vector fields yield new geometric estimates generalising certain results of Lima et al. (Nonlinear Anal. 2010;72:771–781). We also obtain a lower bound on the spectrum of the weighted p-Laplacian on a complete noncompact smooth metric space with the underlying space being a submanifold with bounded mean curvature in the hyperbolic space form of constant negative sectional curvature generalising results of Du and Mao (J Math Anal Appl. 2017;456:787–795).

摘要

本文通过使用满足正发散条件的矢量场，对光滑度量度量空间和具有负截面曲率的子流形的背景进行了概括，给出了一些对p-基调的著名几何估计。选择矢量场是广义麦基恩估计而矢量场的其它选择适当选择的距离函数的产率的梯度产生新的几何估计要概括Lima等的一定的成果。（非线性分析，2010； 72：771-781）。我们还获得了加权p谱的下界-在完全非紧致光滑度量空间上的拉普拉斯算子，其中基础空间是具有恒定负截面曲率的双曲空间形式中有界平均曲率的子流形，并产生了Du和Mao的结果（J Math Anal Appl。2017; 456：787–795） 。