Search results for "Differential geometry"
showing 10 items of 462 documents
Inverse problems for elliptic equations with fractional power type nonlinearities
2020
We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.
Non-parametric mean curvature flow with prescribed contact angle in Riemannian products
2020
Assuming that there exists a translating soliton $u_\infty$ with speed $C$ in a domain $\Omega$ and with prescribed contact angle on $\partial\Omega$, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to $u_\infty +Ct$ as $t\to\infty$. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of $\Omega$ and Ricci curvature in $\Omega$.
A Cornucopia of Carnot groups in Low Dimensions
2022
Abstract Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts. To understand their algebraic structure, it is useful to study some examples explicitly. In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invaria…
On Radon Transforms on Tori
2014
We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvi\`ere. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensors to have distribution valued coefficients. These imply new injectivity results for the periodic broken ray transform on cubes of any dimension.
Invariant distributions, Beurling transforms and tensor tomography in higher dimensions
2014
In the recent articles \cite{PSU1,PSU3}, 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 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 natur…
The Riemannian manifold of all Riemannian metrics
1991
In this paper we study the geometry of (M, G) by using the ideas developed in [Michor, 1980]. With that differentiable structure on M it is possible to use variational principles and so we start in section 2 by computing geodesics as the curves in M minimizing the energy functional. From the geodesic equation, the covariant derivative of the Levi-Civita connection can be obtained, and that provides a direct method for computing the curvature of the manifold. Christoffel symbol and curvature turn out to be pointwise in M and so, although the mappings involved in the definition of the Ricci tensor and the scalar curvature have no trace, in our case we can define the concepts of ”Ricci like cu…
Optimal transport maps on Alexandrov spaces revisited
2018
We give an alternative proof for the fact that in $n$-dimensional Alexandrov spaces with curvature bounded below there exists a unique optimal transport plan from any purely $(n-1)$-unrectifiable starting measure, and that this plan is induced by an optimal map.
Conformal equivalence of visual metrics in pseudoconvex domains
2017
We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between pseudoconvex domains. The proofs are inspired by Mostow's proof of his rigidity theorem and are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.
An index formula on manifolds with fibered cusp ends
2002
We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior and a term that comes from the boundary. This answers the index problem formulated by Mazzeo and Melrose. We give a more precise answer in the case where the base of the boundary fiber bundle is the circle. In particular, for Dirac operators associated to a "product fibered cusp metric", the index is given by the integral of t…
The geometry of canal surfaces and the length of curves in de Sitter space
2011
Abstract We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in correspondence with closed almost-regular canal surfaces, and their length is a natural magnitude in conformal geometry. As an application, we get a lower bound for the total conformal torsion of closed space curves.