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Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)$
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Christopher D. Sogge

Print publication date: 2014

Print ISBN-13: 9780691160757

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691160757.001.0001

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Geodesics and the Hadamard parametrix

Geodesics and the Hadamard parametrix

(p.16) Chapter Two Geodesics and the Hadamard parametrix
Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Christopher D. Sogge

Princeton University Press

This chapter studies the spectrum of Laplace–Beltrami operators on compact manifolds. It begins by defining a metric on an open subset Ω‎ ⊂ Rn, in order to lift their results to corresponding ones on compact manifolds. The chapter then details some elliptic regularity estimates, before embarking on a brief review of geodesics and normal coordinates. The purpose of this review is to show that, with given a particular Laplace–Beltrami operator and any point y0 in Ω‎, one can choose a natural local coordinate system y = κ‎(x) vanishing at y0 so that the quadratic form associated with the metric takes a special form. To conclude, the chapter turns to the Hadamard parametrix.

Keywords:   Laplace–Beltrami operators, compact manifolds, elliptic regularity estimates, geodesics, normal coordinates, Hadamard parametrix

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