带权流形上的纽曼特征值估计

Neumann Eigenvalue Estimates on Weighted Manifolds

讨论一类光滑紧致带权黎曼流形上的纽曼特征值估计问题,假定这类流形具有光滑边界,边界是凸的,而且流形上的Bakery-Emery Ricci曲率具有正的下界.利用了极大模原理去证明热方程解的梯度估计,然后得到热核上界估计.再利用热核与特征值的关系,得到了特征值的下界估计.

In this paper,we discuss a problem of Neumann eigenvalue estimates on a smooth,compact weighted Riemannian manifold.Suppose that the manifold with smooth boundary has a convex boundary and its Bakery-Emery Ricci curvature is positive.We use the maximum principle to prove the gradient estimate of the solution of the heat equation,then we obtain the upper bound estimate of the heat kernel.Finally we use the relationship of heat kernel and eigenvalue to get the lower bound estimate of eigenvalue.