Zoll Manifolds Research Articles

On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievability

We compute the contact manifold of null geodesics of the family of spacetimes left{ left( mathbb {S}^2times mathbb {S}^1, g_circ -frac{d^2}{c^2}dt^2right) right} _{d,cin mathbb {N}^+text { coprime}}, with g_circ the round metric on mathbb {S}^2 and t the mathbb {S}^1-coordinate. We find that these are the lens spaces L(2c, 1) together with the pushforward of the canonical contact structure on STmathbb {S}^2cong L(2,1) under the natural projection L(2,1)rightarrow L(2c,1). We extend this computation to Ztimes mathbb {S}^1 for Z a Zoll manifold. On the other hand, motivated by these examples, we show how Engel geometry can be used to describe the manifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation. We characterize the three-dimensional contact manifolds that are contactomorphic to the space of null geodesics of a spacetime. The characterization consists in the existence of an overlying Engel manifold with a certain foliation and, in this case, we also retrieve the spacetime.

Local and global theory for the three-wave mixing Schrödinger system with quadratic interactions on Zoll Manifolds

Local and global theory for the three-wave mixing Schrödinger system with quadratic interactions on Zoll Manifolds

Growth of Sobolev norms for abstract linear Schrödinger equations

We prove an abstract theorem giving a \langle t\rangle^\epsilon bound (for all \epsilon > 0 ) on the growth of the Sobolev norms in linear Schrödinger equations of the form \mathrm i \dot \psi = H_0 \psi + V(t)\psi as t \to \infty . The abstract theorem is applied to several cases, including the cases where (i) H_0 is the Laplace operator on a Zoll manifold and V(t) a pseudodifferential operator of order smaller than 2; (ii) H_0 is the (resonant or nonresonant) Harmonic oscillator in mathbb \R^d and V(t) a pseudodifferential operator of order smaller than that of H_0 depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of [MR17].

Local and global well-posedness for a quadratic Schrödinger system on Zoll manifolds

We consider the initial value problem (IVP) associated to a quadratic Schrödinger system{i∂tv±Δgv−v=ϵ1uv¯,t∈R,x∈M,iσ∂tu±Δgu−αu=ϵ22v2,σ>0,α∈R,ϵi∈C(i=1,2),(v(0),u(0))=(v0,u0), posed on a d-dimensional compact Zoll manifold M. Considering σ=θβ with θ,β∈{n2:n∈Z} we derive a bilinear Strichartz type estimate and use it to prove the local well-posedness results for given data (v0,u0)∈Hs(M)×Hs(M) whenever s>14 when d=2 and s>d−22 when d≥3. Moreover, in dimensions 2 and 3, we use a Gagliardo-Nirenberg type inequality and conservation laws to prove that the local solution can be extended globally in time whenever s≥1.

Reducibility of Schrödinger equation on a Zoll manifold with unbounded potential

In this article, we prove a reducibility result for the linear Schrödinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less than or equal to 1/2. As far as we know, this is the first reducibility result for an unbounded perturbation on a compact manifold different from the torus.

On time dependent Schrödinger equations: Global well-posedness and growth of Sobolev norms

In this paper we consider time dependent Schrödinger linear PDEs of the form i∂tψ=L(t)ψ, where L(t) is a continuous family of self-adjoint operators. We give conditions for well-posedness and polynomial growth for the evolution in abstract Sobolev spaces.If L(t)=H+V(t) where V(t) is a perturbation smooth in time and H is a self-adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing, we prove that the Sobolev norms of the solution grow at most as tϵ when t↦∞, for any ϵ>0. If V(t) is analytic in time we improve the bound to (log⁡t)γ, for some γ>0. The proof follows the strategy, due to Howland, Joye and Nenciu, of the adiabatic approximation of the flow. We recover most of known results and obtain new estimates for several models including 1-degree of freedom Schrödinger operators on R and Schrödinger operators on Zoll manifolds.

Local well‐posedness of critical nonlinear Schrödinger equation on Zoll manifolds of odd‐growth

In this paper, we study the nonlinear Schrödinger equation on Zoll manifolds with odd order nonlinearities. We will obtain the local well‐poesdness in the critical space . This extends the recent results in the literature to the Zoll manifolds of dimension d≥2 with general odd order nonlinearities and also partially improves the previous results in the subcritical spaces of Yang to the critical cases. Copyright © 2015 John Wiley & Sons, Ltd.

Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

We study the long time dynamics of the Schrodinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrodinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrodinger operators: we prove that adding a potential \({V \in C^{\infty} (\mathbb{S}^{d})}\) to the Laplacian on the sphere results in the existence of geodesics \({\gamma}\) such that the uniform measure supported on \({\gamma}\) cannot be obtained as a weak-\({\star}\) accumulation point of the densities \({(|\psi_{n}|^{2} {vol}_{\mathbb{S}^d})}\) for any sequence of eigenfunctions \({(\psi_n)}\) of \({\Delta_{\mathbb{S}^{d}} - V}\). We also show that the same phenomenon occurs for the free Laplacian on certain Zoll surfaces.

Nonlinear Schrödinger equations on compact Zoll manifolds with odd growth

We study nonlinear Schrodinger equations on Zoll manifolds with nonlinear growth of the odd order. It is proved that local uniform well-posedness are valid in the Hs-subcritical setting according to the scaling invariance, apart from the cubic growth in dimension two. This extends the results by Burq et al. (2005) to higher dimensions with general nonlinearities.

The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds

Let $(M,g)$ be a three-dimensional smooth compact Riemannian manifold such that all geodesics are simple and closed with a common minimal period, such as the $3$-sphere ${\Bbb S}^3$ with canonical metric. In this work the global well-posedness problem for the quintic nonlinear Schr\"odinger equation $i\partial_t u+\Delta u= \pm |u|^4u$, $u|_{t=0}=u_0$ is solved for small initial data $u_0$ in the energy space $H^1(M)$, which is the scaling-critical space. Further, local well-posedness for large data, as well as persistence of higher initial Sobolev regularity is obtained. This extends previous results of Burq-G\'erard-Tzvetkov to the endpoint case.

Existence en temps grand pour des équations de Klein–Gordon à petite donnée initiale sur une structure de Toeplitz

We prove a long time existence, of order C(r)ε−r for all r⩾3, for small solutions, of order ε≪1, in high Sobolev norms of Klein–Gordon equation with Hamiltonian nonlinearities. Manifolds studied are endowed with Toeplitz structures in the sense of Boutet de Monvel and Guillemin. We also make a geometric assumption about periodicity of the Toeplitz pseudo-differential operator Hamiltonian flow. This ensures a useful spectral localization. Our approach follows Delort and Szeftelʼs and Bambusi, Delort, Grebert and Szeftelʼs works on the spheres and Zoll manifolds and uses Birkhoff normal forms at any order. The context of Toeplitz structures allows us to generalize all the previous cases (torus, spheres and Zoll manifolds) and to deal with new linearities involving Szegö projectors.

An abstract Nash–Moser theorem with parameters and applications to PDEs

We prove an abstract Nash–Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the “tame” estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large “clusters of small divisors”, due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity.

Semiclassical measures and the Schrödinger flow on Riemannian manifolds

In this paper we study limits of Wigner distributions (the so-called semiclassical measures) corresponding to sequences of solutions to the semiclassical Schrödinger equation at time scales αh tending to infinity as the semiclassical parameter h tends to zero (when αh = 1/h this is equivalent to considering solutions to the non-semiclassical Schrödinger equation). Some general results are presented, among which is a weak version of Egorov's theorem that holds in this setting. A complete characterization is given for the Euclidean space and Zoll manifolds (that is, manifolds with periodic geodesic flow) via averaging formulae relating the semiclassical measures corresponding to the evolution to those of the initial states. The case of the flat torus is also addressed; it is shown that non-classical behaviour may occur when energy concentrates on resonant frequencies. Moreover, we present an example showing that the semiclassical measures associated with a sequence of states no longer determines those of their evolutions. Finally, some results concerning the equation with a potential are presented.

Some Remarks on Quantum Limits on Zoll Manifolds

We study the concentration of eigenfunctions of the Laplace–Beltrami operator on manifolds all whose geodesics are closed (the so-called Zoll manifolds). Some results on the structure of the set of invariant semiclassical measures associated to sequences of eigenfunctions are given. Among these, we show that any probability measure on the unit tangent bundle of a compact rank-one symmetric space that is invariant by the geodesic flow may be realized as the semiclassical measure of a sequence of eigenfunctions of the Laplacian. This extends a previous result of Jakobson and Zelditch on spheres.

Determinants of zeroth order operators

For compact Riemannian manifolds all of whose geodesics are closed (aka Zoll manifolds) one can define the determinant of a zeroth order pseudodifferential operator by mimicking Szego’s definition of this determinant for the operator: multiplication by a bounded function, on the Hilbert space of square-integrable functions on the circle. In this paper we prove that the non-local contribution to this determinant can be computed in terms of a much simpler “zeta-regularized” determinant.

Almost global existence for Hamiltonian semilinear Klein‐Gordon equations with small Cauchy data on Zoll manifolds

AbstractThis paper is devoted to the proof of almost global existence results for Klein‐Gordon equations on Zoll manifolds (e.g., spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the Laplacian perturbed by a potential on Zoll manifolds. © 2007 Wiley Periodicals, Inc.

Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds

We prove a long time existence result for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds. This generalizes a preceding result concerning the case of spheres, obtained in an earlier paper by the authors. The proof relies on almost orthogonality properties of products of eigenfunctions of positive elliptic selfadjoint operators on a compact manifold and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.

Global existence for nonlinear Klein–Gordon equations in infinite homogeneous waveguides in two dimensions

In this paper we prove a global existence result for nonlinear Klein–Gordon equations with small data in infinite homogeneous waveguids, R 2 × M , where M = ( M , g ) is a Zoll manifold. The method is based on the normal forms, the eigenfunction expansion for M and the special distribution of eigenvalues of Laplace–Beltrami on Zoll manifold.

Global solutions for nonlinear Klein–Gordon equations in infinite homogeneous waveguides

In this paper we prove a global existence result for nonlinear Klein–Gordon equations in infinite homogeneous waveguides, R × M , with smooth small data, where M = ( M , g ) is a Zoll manifold, or a compact revolution hypersurface. The method is based on normal forms, eigenfunction expansion and the special distribution of eigenvalues of the Laplace–Beltrami on such manifolds.

The Schrödinger equation on spheres

It is shown that the fundamental solution to the Schrodinger equation on a d-dimensional sphere has an explicit description at times that are rational multiples of π. This leads to sharp L p estimates on the solution operator at those times. Analogous, though less explicit, results are obtained when spheres are replaced by Zoll manifolds, and when potentials are added.