Publications

2018

• An application of the Duistertmaat--Heckman Theorem and its extensions in Sasaki Geometry
  
  • Boyer Charles P.
  • Huang Hongnian
  • Legendre Eveline
  Geometry and Topology, Mathematical Sciences Publishers, 2018. Building on an idea laid out by Martelli--Sparks--Yau, we use the Duistermaat-Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein--Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Studying the leading terms we prove they are all proper. Among consequences we get that the Einstein-Hilbert functional attains its minimal value and each Sasaki cone possess at least one Reeb vector field with vanishing transverse Futaki invariant. (10.2140/gt.2018.22.4205)
  DOI : 10.2140/gt.2018.22.4205
• HIGH-FREQUENCY BACKREACTION FOR THE EINSTEIN EQUATIONS UNDER POLARIZED U(1) SYMMETRY
  
  • Huneau Cécile
  • Luk Jonathan
  Duke Mathematical Journal, Duke University Press, 2018, 167 (18). Known examples in plane symmetry or Gowdy symmetry show that given a 1-parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a non-trivial stress-energy-momentum tensor. We consider this phenomenon under polarized U(1) symmetry – a much weaker symmetry than most of the known examples – such that the stress-energy-momentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic local-in-time small-data polarized-U(1)-symmetric solution to the Einstein–multiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large, and appears to be the first construction of such examples with more than two families. (10.1215/00127094-2018-0035)
  DOI : 10.1215/00127094-2018-0035
• Dynamical pairs with an absolutely continuous bifurcation measure
  
  • Gauthier Thomas
  , 2018. In this article, we study algebraic dynamical pairs $(f,a)$ parametrized by an irreducible quasi-projective curve $\Lambda$ having an absolutely continuous bifurcation measure. We prove that, if $f$ is non-isotrivial and $(f,a)$ is unstable, this is equivalent to the fact that $f$ is a family of Latt\`es maps. To do so, we prove the density of transversely prerepelling parameters in the bifucation locus of $(f,a)$ and a similarity property, at any transversely prerepelling parameter $\lambda_0$, between the measure $\mu_{f,a}$ and the maximal entropy measure of $f_{\lambda_0}$. We also establish an equivalent result for dynamical pairs of $\mathbb{P}^k$, under an additional assumption.
• Levi-Kahler reduction of CR structures, products of spheres, and toric geometry
  
  • Legendre Eveline
  • Apostolov Vestislav
  • Calderbank David M J
  • Gauduchon Paul
  , 2018. We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of such quotients, and find Levi-Kahler quotients of products of 3-spheres which are extremal in a weighted sense introduced by G. Maschler and the first author.
• Toric contact geometry in arbitrary codimension
  
  • Apostolov Vestislav
  • Calderbank David M J
  • Gauduchon Paul
  • Legendre Eveline
  , 2018. We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.
• A MEAN-FIELD LIMIT OF THE LOHE MATRIX MODEL AND EMERGENT DYNAMICS
  
  • Golse François
  • Ha Seung-Yeal
  , 2018. The Lohe matrix model is a continuous-time dynamical system describing the collective dynamics of group elements in the unitary group man-ifold, and it has been introduced as a toy model of a non abelian generalization of the Kuramoto phase model. In the absence of couplings, it reduces to the finite-dimensional decoupled free Schrödinger equations with constant Hamiltonians. In this paper, we study a rigorous mean-field limit of the Lohe matrix model which results in a Vlasov type equation for the probability density function on the corresponding phase space. We also provide two different settings for the emergent synchronous dynamics of the kinetic Lohe equation in terms of the initial data and the coupling strength.
• Two-bubble dynamics for threshold solutions to the wave maps equation
  
  • Jendrej Jacek
  • Lawrie Andrew
  Inventiones Mathematicae, Springer Verlag, 2018, 213 (3), pp.1249-1325. (10.1007/s00222-018-0804-2)
  DOI : 10.1007/s00222-018-0804-2
• Degeneration of endomorphisms of the complex projective space in the hybrid space
  
  • Favre Charles
  Journal of the Institute of Mathematics of Jussieu, Cambridge University Press, 2018. We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk. We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holds in the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms. (10.1017/S147474801800035X)
  DOI : 10.1017/S147474801800035X
• Mixed Hodge modules without slope
  
  • Kochersperger Matthieu
  , 2018. In this article we are interested in morphisms without slope for mixed Hodge modules. We first show the commutativity of iterated nearby cycles and vanishing cycles applied to a mixed Hodge module in the case of a morphism without slope. Then we define the notion "strictly without slope" for a mixed Hodge module and we show the preservation of this condition under the direct image by a proper morphism. As an application we prove the compatibility of the Hodge filtration and Kashiwara-Malgrange filtrations for some pure Hodge modules with support an hypersurface with quasi-ordinary singularities.
• On propagation of higher space regularity for non-linear Vlasov equations
  
  • Han-Kwan Daniel
  Analysis & PDE, Mathematical Sciences Publishers, 2018. This work is concerned with the broad question of propagation of regularity for smooth solutions to non-linear Vlasov equations. For a class of equations (that includes Vlasov-Poisson and relativistic Vlasov-Maxwell), we prove that higher regularity in space is propagated, locally in time, into higher regularity for the moments in velocity of the solution. This in turn can be translated into some anisotropic Sobolev higher regularity for the solution itself, which can be interpreted as a kind of weak propagation of space regularity. To this end, we adapt the methods introduced in the context of the quasineutral limit of the Vlasov-Poisson system in [D. Han-Kwan and F. Rousset, Ann. Sci. \'Ecole Norm. Sup., 2016]. (10.2140/apde.2019.12.189)
  DOI : 10.2140/apde.2019.12.189
• INTERACTION OF SOLITONS FROM THE PDE POINT OF VIEW
  
  • Martel Yvan
  , 2018. We review recent results concerning the interactions of solitary waves for several universal nonlinear dispersive or wave equations. Though using quite different techniques, these results are partly inspired by classical papers based on the inverse scattering theory for integrable models.
• Variational and non-Archimedean aspects of the Yau-Tian-Donaldson conjecture
  
  • Boucksom Sébastien
  , 2018, pp.591-617. (10.1142/9789813272880_0069)
  DOI : 10.1142/9789813272880_0069
• LINEAR BOLTZMANN EQUATION AND FRACTIONAL DIFFUSION
  
  • Bardos Claude
  • Golse François
  • Moyano Iván
  Kinetic and Related Models, AIMS, 2018, 11 (4), pp.1011-1036. Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient σ. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient α. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where σ → +∞ and 1 − α=C/σ, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of a kinetic model which is based on the harmonic extension definition of √ −∆. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions (" heavy tails ") or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273–280]. (10.3934/krm.2018039)
  DOI : 10.3934/krm.2018039
• Synthesis of Fluorescent BODIPY-Labeled Analogue of Miltefosine for Staining of Acanthamoeba .
  
  • Courrier Emilie
  • Maret Corentin
  • Charaoui-Boukerzaza Sana
  • Lambert Victor
  • de Nicola Antoinette
  • Muzuzu Wenziz
  • Ulrich Gilles
  • Raberin Hélène
  • Flori Pierre
  • Moine Baptiste
  • He Zhiguo
  • Gain Philippe
  • Thuret Gilles
  ChemistrySelect, Wiley, 2018, 3 (27), pp.7674 - 7679. (10.1002/slct.201801159)
  DOI : 10.1002/slct.201801159
• Croissance des degrés d'applications rationnelles en dimension 3
  
  • Dang Nguyen-Bac
  , 2018. Cette thèse comporte trois chapitres indépendants portant sur l’itération des applicationsrationnelles sur des variétés projectives et plus spécifiquement sur l’étude du comportement dela suite des degrés des itérés de telles applications.Dans le premier chapitre, nous donnons une construction des invariants fondamentaux quesont les degrés dynamiques dans un cadre très général, et ce sans hypothèse ni sur la caractéristique ni sur les singularités de l’espace ambiant. Cette construction repose sur des propriétésde positivité des cycles algébriques, et propose une alternative aux approches analytiques deDinh et Sibony ou algébriques de Truong.Le second chapitre est issu d’un article écrit en commun avec Jian Xiao. Notre contributionporte sur des objets centraux en géométrie convexe appelés valuations. Nous transférons à l’espace des valuations des notions de positivité des cycles algébriques récemment introduites parLehmann et Xiao, ce qui nous permet d’étendre l’opération de convolution originellement définie par Bernig et Fu à une sous-classe de valuations suffisamment positives.Le troisième chapitre constitue le coeur de la thèse, et porte sur des estimations des degrésdynamiques des automorphismes dit modérés de la quadrique affine de dimension 3. Nos arguments sont de nature variée, et s’appuient sur l’action du groupe modéré sur un complexe carréCAT(0) et Gromov hyperbolique récemment introduite par Bisi, Furter et Lamy.Nous avons finalement collecté dans un dernier et court chapitre quelques pistes de recherchedirectement inspirées des travaux présentés ici.
• Long time estimates for the Vlasov-Maxwell system in the non-relativistic limit
  
  • Han-Kwan Daniel
  • Nguyen Toan T.
  • Rousset Frédéric
  Communications in Mathematical Physics, Springer Verlag, 2018. In this paper, we study the Vlasov-Maxwell system in the non-relativistic limit, that is in the regime where the speed of light is a very large parameter. We consider data lying in the vicinity of homogeneous equilibria that are stable in the sense of Penrose (for the Vlasov-Poisson system), and prove Sobolev stability estimates that are valid for times which are polynomial in terms of the speed of light and of the inverse of size of initial perturbations. We build a kind of higher-order Vlasov-Darwin approximation which allows us to reach arbitrarily large powers of the speed of light. (10.1007/s00220-018-3208-7)
  DOI : 10.1007/s00220-018-3208-7
• On some constructions of contact manifolds
  
  • Gironella Fabio
  , 2018. This thesis is divided in two parts.The first part focuses on the study of the topology of the contactomorphism group of some explicit high dimensional contact manifolds. More precisely, using constructions and results by Massot, Niederkrüger and Wendl, we construct (infinitely many) examples in all dimensions of contactomor-phisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopicto the identity. We also give examples of tight high dimensional contact manifolds admitting a contactomorphism whose powers are all smoothly isotopic but not contact-isotopic to the identity ;this is a generalization of a result in dimension 3 by Ding and Geiges.In the second part, we construct examples of higher dimensional contact manifolds with specific properties. This leads us to the existence of tight virtually overtwisted closed contact manifolds in all dimensions and to the fact that every closed contact 3-manifold embeds with trivial nor-mal bundle inside a tight closed contact 5-manifold. This uses known construction procedures byBourgeois (on products with tori) and Geiges (on branched covering spaces). We pass from these procedures to definitions ; this allows to prove a uniqueness statement in the case of contact branched coverings, and to study the global properties (such as tightness and fillability) of the results of both constructions without relying on any auxiliary choice in the procedures. A second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book, due to Giroux, and of contact fiber bundle, due to Lerman. For instance,we give a definition of Bourgeois contact structures on flat contact fiber bundles which is local,(strictly) includes the results of Bourgeois’ construction, and allows to recover an isotopy class of supporting open books on the fibers. This last point relies on a reinterpretation, inspired by anidea by Giroux, of supporting open books in terms of pairs of contact vector fields.
• Convolution intermédiaire et théorie de Hodge
  
  • Martin Nicolas
  , 2018. Cette thèse est constituée de deux parties complètement indépendantes.Dans une première partie, nous montrons que la paire de Fourier-Mukai (X,Y) issue de la correspondance double miroir Pfaffienne-Grassmannienne vérifie l'identité ([X]-[Y])L^6=0 dans l'anneau de Grothendieck, où L est la classe de la droite affine. Ce résultat est un raffinement d'un théorème de Borisov par la suppression d'un facteur, qui montre que la classe de la droite affine est un diviseur de zéro dans l'anneau de Grothendieck, et fournit par ailleurs un premier exemple intéressant de variétés D-équivalentes qui sont L-équivalentes. D'autres exemples ont par la suite été explicités par d'autres auteurs.Dans une seconde partie, nous nous intéressons au comportement d'invariants de théorie de Hodge par convolution intermédiaire, à la suite des travaux de Dettweiler et Sabbah. Le principal résultat concerne le comportement des données numériques locales de Hodge cycles proches à l'infini par convolution intermédiaire additive par un module de Kummer. Nous donnons également des formules pour les invariants locaux h^p et globaux delta^p sans faire l'hypothèse de monodromie scalaire à l'infini. De plus, à l'aide d'une relation de Katz reliant les convolutions additives et multiplicatives, nous explicitons le comportement des invariants de Hodge par convolution intermédiaire multiplicative. Enfin, le théorème principal permet de redémontrer un résultat de Fedorov sur les invariants de Hodge d'équations hypergéométriques.
• Cycles proches, cycles évanescents et théorie de Hodge pour les morphismes sans pente
  
  • Kochersperger Matthieu
  , 2018. Dans cette thèse nous nous intéressons aux singularités d'espaces analytiques complexes définis comme le lieu des zéros d'un morphisme sans pente. Nous étudions dans un premier temps les cycles proches et les cycles évanescents associés à un tel morphisme. Dans un deuxième temps nous cherchons à comprendre la théorie de Hodge des morphismes sans pente.La première partie de cette thèse est consacrée à apporter des compléments au travail de P. Maisonobe sur les morphismes sans pente. Nous commençons par construire un morphisme de comparaison entre cycles proches algébriques (pour les D-modules) et cycles proches topologiques (pour les faisceaux pervers). Nous montrons ensuite que ce morphisme est un isomorphisme dans le cas d'un morphisme sans pente. Enfin nous construisons un foncteur cycles évanescents topologiques pour un morphisme sans pente et nous démontrons que ce foncteur et le foncteur cycles proches topologiques de P. Maisonobe se placent dans le diagramme de triangles exacts attendu.Dans la seconde partie de cette thèse nous étudions les morphismes sans pente pour les modules de Hodge mixtes. Nous démontrons dans un premier temps la commutativité des cycles proches et des cycles évanescents itérés appliqués à un module de Hodge mixte dans le cas d'un morphisme sans pente. Dans un deuxième temps nous définissons la notion "strictement sans pente" pour un module de Hodge mixte et nous démontrons sa stabilité par image directe propre. Nous démontrons comme application la compatibilité de la filtration de Hodge et des filtrations de Kashiwara-Malgrange pour certains modules de Hodge purs supportés sur une hypersurface à singularités quasi-ordinaires.
• EINSTEIN EQUATIONS UNDER POLARIZED U(1) SYMMETRY IN AN ELLIPTIC GAUGE
  
  • Huneau Cécile
  • Luk Jonathan
  Communications in Mathematical Physics, Springer Verlag, 2018, 361 (3), pp.873-949. We prove local existence of solutions to the Einstein–null dust system under polarized U(1) symmetry in an elliptic gauge. Using in particular the previous work of the first author on the constraint equations, we show that one can identify freely prescribable data, solve the constraints equations, and construct a unique local in time solution in an elliptic gauge. Our main motivation for this work, in addition to merely constructing solutions in an elliptic gauge, is to provide a setup for our companion paper in which we study high frequency backreaction for the Einstein equations. In that work, the elliptic gauge we consider here plays a crucial role to handle high frequency terms in the equations. The main technical difficulty in the present paper, in view of the application in our companion paper, is that we need to build a framework consistent with the solution being high frequency, and therefore having large higher order norms. This difficulty is handled by exploiting a reductive structure in the system of equations. (10.1007/s00220-018-3167-z)
  DOI : 10.1007/s00220-018-3167-z
• STRONGLY INTERACTING BLOW UP BUBBLES FOR THE MASS CRITICAL NLS
  
  • Martel Yvan
  • Raphaël Pierre
  , 2018. We consider the mass critical two dimensional nonlinear Schrödinger equation (NLS) i∂tu + ∆u + |u| 2 u = 0, t ∈ R, x ∈ R 2. Let Q denote the positive ground state solitary wave satisfying ∆Q − Q + Q 3 = 0. We construct a new class of multi–solitary wave solutions: given any integer K ≥ 2, there exists a global (for t > 0) solution u(t) of (NLS) that decomposes asymptotically into a sum of solitary waves centered at the vertices of a K–sided regular polygon and concentrating at a logarithmic rate as t → +∞ so that the solution blows up in infinite time with the rate ∇u(t) L 2 ∼ | log t| as t → +∞. This special behavior is due to strong interactions between the waves, in contrast with previous works on multi–solitary waves of (NLS) where interactions do not affect the blow up rate. Using the pseudo–conformal symmetry of the (NLS) flow, this yields the first example of solution v(t) of (NLS) blowing up in finite time with a rate strictly above the pseudo–conformal one, namely, ∇v(t) L 2 ∼ log |t| t as t ↑ 0. Such solution concentrates K bubbles at a point x0 ∈ R 2 , i.e. |v(t)| 2 ⇀ KQ 2 L 2 δx 0 as t ↑ 0.
• Stability of Minkowski Space-time with a translation space-like Killing eld
  
  • Huneau Cécile
  Annals of PDE, Springer, 2018, 4 (1), pp.12. In this paper we prove the nonlinear stability of Minkowski space-time with a translation Killing eld. In the presence of such a symmetry, the 3 + 1 vacuum Einstein equations reduce to the 2 + 1 Einstein equations with a scalar eld. We work in generalised wave coordinates. In this gauge Einstein's equations can be written as a system of quasilinear quadratic wave equations. The main diculty in this paper is due to the decay in 1 √ t of free solutions to the wave equation in 2 dimensions, which is weaker than in 3 dimensions. This weak decay seems to be an obstruction for proving a stability result in the usual wave coordinates. In this paper we construct a suitable generalized wave gauge in which our system has a "cubic weak null structure", which allows for the proof of global existence. 1 Introduction In this paper, we address the stability of the Minkowski solution to the Einstein vacuum equations with a translation space-like Killing eld. More precisely, we look for solutions of the 3 + 1 vacuum Einstein equations, on manifolds of the form Σ × R x 3 × R t , where Σ is a 2 dimensional manifold, equipped with a metric of the form g = e −2φ g + e 2φ (dx 3) 2 , where φ is a scalar function, and g a Lorentzian metric on Σ × R, all quantities being independent of x 3. For these metrics, Einstein vacuum equations are equivalent to the 2 + 1 dimensional system g φ = 0 R µν = 2∂ µ φ∂ ν φ, (1.1) where R µν is the Ricci tensor associated to g. Choquet-Bruhat and Moncrief studied the case where Σ is compact of genus G ≥ 2. In [7], they proved the stability of a particular expanding solution. In this paper we work in the case Σ = R 2. A particular solution is then given by Minkowski solution itself. It corresponds to φ = 0 and g = m, the Minkowski metric in dimension 2 + 1. A natural question one can ask in this setting is the nonlinear stability of this solution. In the 3+1 vacuum case, the stability of Minkowski space-time has been proven in the celebrated work of Christodoulou and Klainerman [8] in a maximal foliation. It has then been proven by Lindblad and Rodnianski using wave coordinates in [23]. Their proof extends also to Einstein equations coupled to a scalar eld. Let us note that the perturbations of Minkowski solution considered in our paper are not asymptotically at in 3 + 1 dimension, due to the presence of a translation Killing eld. Consequently they are not included in [8] and [23]. (10.1007/s40818-018-0048-x)
  DOI : 10.1007/s40818-018-0048-x
• Sur les paquets d'Arthur des groupes unitaires et quelques cons\'equences pour les groupes classiques
  
  • Renard David
  • Moeglin Colette
  , 2018. We give an explicit construction of Arthur packets for real unitary groups by cohomological and parabolic induction and following an idea communicated to us by P. Trapa, we show that they satisfy the multiplicity one property. In particular, we show the irreducibility of some parabolically induced representations for unitary groups, and use this to give the proof of analogous statements made in our work on Arthur packets of classical groups. Nous donnons une construction explicite des paquets d'Arthur des groupes unitaires r\'eels par induction cohomologique et induction parabolique et en suivant une id\'ee communiqu\'ee par P. Trapa, nous \'etablissons la propri\'et\'e de multiplicit\'e un de ceux-ci. Nous montrons en particulier des r\'esultats d'irr\'eductibilit\'e de certaines induites paraboliques pour les groupes unitaires, ce qui nous permet de compl\'eter les d\'emonstrations d'\'enonc\'es analogues annonc\'es dans nos travaux sur les paquets d'Arthur des groupes classiques.
• Sur les paquets d'Arthur des groupes classiques et unitaires non quasi-d\'eploy\'es
  
  • Renard David
  • Moeglin Colette
  , 2018. Nous \'etendons aux groupes orthogonaux et unitaires non quasi-d\'eploy\'es sur un corps local des r\'esultats de J. Arthur et de la premi\`ere auteure \'etablis dans le cas quasi-d\'eploy\'e. En particulier, nous obtenons une classification de Langlands compl\`ete pour les repr\'esentations temp\'er\'ees dans le cas $p$-adique. Nous en d\'eduisons en utilisant l'involution d'Aubert-Schneider-Stuhler un r\'esultat de multiplicit\'e un dans les paquets unipotents, et par des m\'ethodes globales, le m\^eme r\'esultat pour les paquets unipotents dans le cas archim\'edien. We extend to non quasi-split orthogonal and unitary groups over a local field some results of J. Arthur and the first author established in the quasi-split case. In particular, we obtain a full Langlands classification for tempered representations in the $p$-adic case. Using Aubert-Schneider-Stuhler involution, we deduce from this a multiplicity one result for unipotent packets, and by global methods, the same result for unipotent packets in the archimedean case.
• The Camassa-Holm equation as an incompressible Euler equation: a geometric point of view
  
  • Gallouët Thomas
  • Vialard François-Xavier
  Journal of Differential Equations, Elsevier, 2018, 264 (7). The group of diffeomorphisms of a compact manifold endowed with the L^2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon measures where the Wasserstein-Fisher-Rao distance is a natural extension of the classical L^2-Wasserstein distance. In this paper, we show a similar relation between this unbalanced optimal transport problem and the Hdiv right-invariant metric on the group of diffeomorphisms, which corresponds to the Camassa-Holm (CH) equation in one dimension. On the optimal transport side, we prove a polar factorization theorem on the automorphism group of half-densities. Geometrically, our point of view provides an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L^2 type of cone metric. This leads to a new formulation of the (generalized) CH equation as a geodesic equation on an isotropy subgroup of this automorphisms group; On S1, solutions to the standard CH thus give particular solutions of the incompressible Euler equation on a group of homeomorphisms of R^2 which preserve a radial density that has a singularity at 0. An other application consists in proving that smooth solutions of the Euler-Arnold equation for the Hdiv right-invariant metric are length minimizing geodesics for sufficiently short times. (10.1016/j.jde.2017.12.008)
  DOI : 10.1016/j.jde.2017.12.008