Caltech/UCLA Joint Analysis Seminar

We consider the problem of determining optimal upper bounds on the growth of L^p norms of eigenfunctions of the Laplacian on a compact Riemannian manifold in the high frequency limit. After an introduction to the problem, we will discuss results relating such upper bounds to mass concentration estimates in tubes about geodesic segments with a frequency dependent diameter. When the manifold has nonpositive sectional curvatures, it can be shown that there is a logarithmic gain in the trivial mass concentration estimates over these tubes, translating to improvements in the L^p bounds. These are results in joint works with C. Sogge.