The resolvent of the Laplacian on non-trapping asymptotically hyperbolic manifolds

Yiran Wang, Purdue University

Abstract

Asymptotically hyperbolic manifolds (AHM) are natural generalizations of the hyperbolic space. The spectral and scattering theory for AHM has been studied in connection to conformal geometry and problems in mathematical physics. In this work, we consider non-trapping AHM and study the meromorphic properties and L2 estimates for the resolvent of the Laplacian as well as the asymptotics of solutions to wave equations. We prove that for non-trapping AHM, the resolvent of the Laplacian has a holomorphic continuation to strip-type regions above the real axis in the complex plane where we obtain the resolvent estimates uniformly for large spectral parameters. The key part of the proof and the main contribution of the thesis is to construct an approximation of the resolvent, called a parametrix of the Laplacian, which is uniformly for large energies. To understand the asymptotic properties of the parametrix, we analyze the global behavior of the geodesic flows on non-trapping AHM. As another application of the parametrix, we prove the expansion of solutions to linear wave equations in terms of resonances, as well as the exponential decay of the Frielander radiation fields.

Degree

Ph.D.

Advisors

Sa Barreto, Purdue University.

Subject Area

Mathematics

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