by Joonas Ilmavirta and Antti Kykkänen
Abstract:
We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.
Reference:
Tensor tomography on negatively curved manifolds of low regularity (Joonas Ilmavirta and Antti Kykkänen), Journal of Geometric Analysis, volume 34, pp. 146–187, 2024.
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We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.
[arXiv]
Bibtex Entry:
@article{c11-tensor-tomography,
author = {Joonas Ilmavirta and Antti Kykk\"{a}nen},
title = {{Tensor tomography on negatively curved manifolds of low regularity}},
month = mar,
year = 2024,
arxiv = {2303.08208},
journal = {Journal of Geometric Analysis},
volume = 34,
issue 147,
pages = {146--187},
url={http://users.jyu.fi/~jojapeil/pub/c11-tensor-tomography.pdf},
abstract = {We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with $C^{1,1}$ metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold.
Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.},
doi = {10.1007/s12220-024-01588-8}
}