具有pinched负曲率的黎曼流形上不存在超指数衰减本征函数
Absence of super-exponentially decaying eigenfunctions on Riemannian manifolds with pinched negative curvature
摘要：
设（X，g）是度量完备的单连通黎曼流形，具有有界几何和收缩负曲率，即存在常数a>b>0使得对所有截面曲率K都有-a^2<K<-b^2.这里使用的有界几何的意义上说，所有的共变导数的黎曼曲率张量有界和注入半径是一致的有界以下由一个正常数。我们证明了g的拉普拉斯算子不存在超指数衰减的本征函数。对其他几何算子也给出了类似的结论，并证明了一个定理。
英文摘要：
Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is used in the sense that all covariant derivatives of the Riemannian curvature tensor are bounded and the injectivity radius is uniformly bounded below by a positive constant. We show that there are no superexponentially decaying eigenfunctions of the Laplacian of g. We also show the analogous conclusion for other geometric operators, and prove a theorem with the assumptions and conclusions localized near infinity.
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