Optimal growth bounds for eigenfunctions

Abstract

Analysis of non-compact manifolds almost always requires some controlled behavior at infinity. Without such, one neither can show, nor expect, strong properties. On the other hand, such assumptions restrict the possible applications and often too severely. In a wide range of areas non-compact spaces come with a Gaussian weight and a drift Laplacian. Eigenfunctions are $L^2$ in the weighted space allowing for extremely rapid growth. Rapid growth would be disastrous for many applications. Surprisingly, for very general tensors, manifolds and weights, we show the same polynomial growth bounds that Laplace and Hermite observed for functions on Euclidean space for the standard Gaussian. This covers all shrinkers for Ricci and mean curvature flows. These results open a door for understanding general non-compact spaces. It provides an analytic framework for doing nonlinear PDE on Gaussian spaces where previously the Gaussian weight allowed wild growth that made it impossible to approximate nonlinear by linear. It is key to bound the growth of diffeomorphisms of non-compact manifolds and is the key for solving the "gauge problem". The relative nature of the estimates and the slow growth in the bounds lead to "propagation of almost splitting" that is significantly stronger than pseudo locality and key for applications.