Invariant distributions, Beurling transforms and tensor tomography in higher dimensions

By Gabriel P. Paternain, Mikko Salo, Gunther Uhlmann. Mathematische Annalen, 2014

71 citations recorded by the source corpus.

In the recent articles Paternain et al. (J. Differ Geom, 98:147–181, 2014, Invent Math 193:229–247, 2013), a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the existence of certain distributions that are invariant under the geodesic flow. We prove that on any Anosov manifold, one can find invariant distributions with controlled first Fourier coefficients. The proof is based on subelliptic type estimates and a Pestov identity. We present an alternative construction valid on manifolds with nonpositive curvature, based on the fact that a natural Beurling transform on such manifolds turns out to be essentially a contraction. Finally, we obtain uniqueness results in tensor tomography both on simple and Anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified Jacobi equation.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/6bc5c40abad6b126a59d561bc9ffa751b6138f95

Citations in the corpus

• Lens rigidity for manifolds with hyperbolic trapped set (2014, unknown): “Each u ∈ L 2 ( SM e ) function can be decomposed using the spectral decomposition of the vertical Laplacian Δ v in the fibers of SM e (which are spheres) where u k are L 2 sections of a vector bundle over M e ; see [GuKa2,PSU2].”
• Invariant distributions and X-ray transform for Anosov flows (2014, unknown): “The result improves the recent work of [PSU2] in terms of regularity.”