Search results for "Euclidean"
showing 10 items of 185 documents
Non-preserved curvature conditions under constrained mean curvature flows
2014
We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.
A rigidity problem on the round sphere
2015
We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.
Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below
2013
We show that in any infinitesimally Hilbertian CD* (K,N)-space at almost every point there exists a Euclidean weak tangent, i.e., there exists a sequence of dilations of the space that converges to Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian CD* (0,N)-spaces.
Tensor tomography on Cartan–Hadamard manifolds
2017
We study the geodesic X-ray transform on Cartan-Hadamard manifolds, and prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature decays at infinity. This work extends the results of Lehtonen (2016) to dimensions $n \geq 3$ and to the case of tensor fields of any order.
X-ray transforms in pseudo-Riemannian geometry
2016
We study the problem of recovering a function on a pseudo-Riemannian manifold from its integrals over all null geodesics in three geometries: pseudo-Riemannian products of Riemannian manifolds, Minkowski spaces and tori. We give proofs of uniqueness anc characterize non-uniqueness in different settings. Reconstruction is sometimes possible if the signature $(n_1,n_2)$ satisfies $n_1\geq1$ and $n_2\geq2$ or vice versa and always when $n_1,n_2\geq2$. The proofs are based on a Pestov identity adapted to null geodesics (product manifolds) and Fourier analysis (other geometries). The problem in a Minkowski space of any signature is a special case of recovering a function in a Euclidean space fro…
The Calderón problem with partial data on manifolds and applications
2013
We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flatness condition include parts of cylindrical sets, conical sets, and surfaces of revolution. We prove local uniqueness in the Calderon problem with partial data in admissible geometries, and global uniqueness under an additional concavity assumption. This work unifies two earlier approaches to this problem (\cite{KSU} and \cite{I}) and extends both. The proofs are based on impr…
Regularity properties of spheres in homogeneous groups
2015
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context of sub-Finsler manifolds with no abnormal geodesics. Secondly, for general groups we identify an alg…
A reflection approach to the broken ray transform
2013
We reduce the broken ray transform on some Riemannian manifolds (with corners) to the geodesic ray transform on another manifold, which is obtained from the original one by reflection. We give examples of this idea and present injectivity results for the broken ray transform using corresponding earlier results for the geodesic ray transform. Examples of manifolds where the broken ray transform is injective include Euclidean cones and parts of the spheres $S^n$. In addition, we introduce the periodic broken ray transform and use the reflection argument to produce examples of manifolds where it is injective. We also give counterexamples to both periodic and nonperiodic cases. The broken ray t…
Algebraic models of the Euclidean plane
2018
We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the euclidean plane, contrary to the compact case.
Regularity of sets with constant horizontal normal in the Engel group
2012
In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respect to the Euclidean distance). However we find that, when they are written as intrinsic horizontal upper-graphs with respect to the direction of the normal, then the function defining the set might even fail to be continuous. Nevertheless, we can prove that one can always find other horizontal directions for which the set is the intrinsic horizontal upper-graph of a function t…