Open Access
August 2014 Spectral rigidity and invariant distributions on Anosov surfaces
Gabriel P. Paternain, Mikko Salo, Gunther Uhlmann
J. Differential Geom. 98(1): 147-181 (August 2014). DOI: 10.4310/jdg/1406137697


This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal tensors. The surjectivity results are of independent interest and imply the existence of many geometric invariant distributions on the unit sphere bundle. In particular, we show that on any Anosov surface $(M, g)$, given a smooth function $f$ on $M$, there is a distribution in the Sobolev space $H^{-1}(SM)$ that is invariant under the geodesic flow and whose projection to $M$ is the given function $f$.


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Gabriel P. Paternain. Mikko Salo. Gunther Uhlmann. "Spectral rigidity and invariant distributions on Anosov surfaces." J. Differential Geom. 98 (1) 147 - 181, August 2014.


Published: August 2014
First available in Project Euclid: 23 July 2014

zbMATH: 1304.37021
MathSciNet: MR3263517
Digital Object Identifier: 10.4310/jdg/1406137697

Rights: Copyright © 2014 Lehigh University

Vol.98 • No. 1 • August 2014
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