We shall explore the role that curvature plays in harmonic analysis on compact manifolds. We shall focus on estimates that measure the concentration of eigenfunctions.  Using them we are able to affirm the classical Bohr correspondence principle and obtain a new classification of compact space forms in terms of the growth rates of various norms of (approximate) eigenfunctions.  This is joint work with Xiaoqi Huang following earlier work with Matthew Blair.

Also, in joint work with Huang, Zhongkai Tao and Zhexing Zhang, we are able to extend these estimates to manifolds of bounded geometry and nonpositive curvature, allowing us to obtain new estimates on convex cocompact hyperbolic surfaces.