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# Matt Blair, University of New Mexico – Analysis & PDE Seminar

## August 28, 2019 @ 4:00 pm - 5:00 pm

**Title:** L^p bounds for eigenfunctions at the critical exponent

**Abstract:** We consider upper bounds on the growth of L^p norms of eigenfunctions of the Laplacian on a compact Riemannian manifold in the high frequency limit. In particular, we seek to identify geometric or dynamical conditions on the manifold which yield improvements on the universal L^p bounds of C. Sogge. The emphasis will be on bounds at the “critical exponent”, where a spectrum of scenarios for phase space concentration must be considered. This turns out to be reminiscent of phenomena in dispersive PDE with critical nonlinearities. We then discuss a recent work with C. Sogge which shows that when the sectional curvatures are nonpositive, there is a logarithmic type gain in the known L^p bounds at the critical exponent.