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Session S33 - Spectral Geometry

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Fourier restriction on hyperbolic manifolds

Xiaolong Han

California State University, Northridge, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The Fourier restriction phenomenon asks whether one can meaningfully restrict the Fourier transform of a function onto a hypersurface (such as a sphere) in the frequency space. Stein’s restriction conjectures state that the Fourier transform of an Lp function restricts to a well-defined Lq function on the sphere, for appropriate ranges of exponents p and q. While the full conjecture remains open, Tomas and Stein in the 1970s proved the case when q=2. Via the spectral measure, the Tomas-Stein restriction estimates have been proved in geometries other than the Euclidean spaces such as asymptotically conic or hyperbolic manifolds, all of which require that there is no geodesic trapping, i.e., all geodesics extend to infinity. In this talk, we study how the restriction estimates are influenced by this trapping condition. We present the first examples of manifolds with geodesic trapping for which the Tomas-Stein restriction estimates hold.

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