Search results for "Riemannin monistot"
showing 6 items of 6 documents
Universal infinitesimal Hilbertianity of sub-Riemannian manifolds
2019
We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all weighted sub-Finsler manifolds. As an intermediate tool, of independent interest, we show that any sub-Finsler distance can be monotonically approximated from below by Finsler ones. All the results are obtained in the general setting of possibly rank-varying structures.
Pestov identities and X-ray tomography on manifolds of low regularity
2021
We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g \in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $C^{1,1}$-regularity is optimal on the H\"older scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure.
Linearized Calderón problem and exponentially accurate quasimodes for analytic manifolds
2022
In this article we study the linearized anisotropic Calderon problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calderon problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial ingredients in the proof of our result are the construction of Gaussian beam quasimodes on the tra…
Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds
2021
We prove that if $M$ is a closed $n$-dimensional Riemannian manifold, $n \ge 3$, with ${\rm Ric}\ge n-1$ and for which the optimal constant in the critical Sobolev inequality equals the one of the $n$-dimensional sphere $\mathbb{S}^n$, then $M$ is isometric to $\mathbb{S}^n$. An almost-rigidity result is also established, saying that if equality is almost achieved, then $M$ is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the ${\rm RCD}$-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds. An independent result of our analysis is the characterization of the best constant in the Sob…
Translating Solitons Over Cartan-Hadamard Manifolds
2020
We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions.
The Calderón problem for the conformal Laplacian
2022
We consider a conformally invariant version of the Calderón problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states that a locally conformally real-analytic manifold in dimensions can be determined in this way, giving a positive answer to an earlier conjecture [LU02, Conjecture 6.3]. The proof proceeds as in the standard Calderón problem on a real-analytic Riemannian manifold, but new features appear due to the conformal structure. In particular, we introduce a new coordinate system that replaces harmonic coordinates when determining the conformal class in a …