Diffeomorphisms Research Articles

On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct explicit quasimorphisms on [Formula: see text] some of them are [Formula: see text]-continuous and vanish on all autonomous diffeomorphisms, and some of them are Calabi.

A Quasi-isometric embedding into the group of Hamiltonian diffeomorphisms with Hofer’s metric

We construct an embedding Φ of [0, 1]∞ into Ham(M, ω), the group of Hamiltonian diffeomorphisms of a suitable closed symplectic manifold (M, ω). We then prove that Φ is in fact a quasi-isometry. After imposing further assumptions on (M, ω), we adapt our methods to construct a similar embedding of ℝ ⊕ [0, 1]∞ into either Ham(M, ω) or , the universal cover of Ham(M, ω). Along the way, we prove results related to the filtered Floer chain complexes of radially symmetric Hamiltonians. Our proofs rely heavily on a continuity result for barcodes (as presented in [28]) associated to filtered Floer homology viewed as a persistence module.

Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models

The Nambu-Goldstone (NG) bosons of the SYK model are described by a coset space Diff/SL(2, ℝ), where Diff, or Virasoro group, is the group of diffeomorphisms of the time coordinate valued on the real line or a circle. It is known that the coadjoint orbit action of Diff naturally turns out to be the two-dimensional quantum gravity action of Polyakov without cosmological constant, in a certain gauge, in an asymptotically flat spacetime. Motivated by this observation, we explore Polyakov action with cosmological constant and boundary terms, and study the possibility of such a two-dimensional quantum gravity model being the AdS dual to the low energy (NG) sector of the SYK model. We find strong evidences for this duality: (a) the bulk action admits an exact family of asymptotically AdS2 spacetimes, parameterized by Diff/SL(2, ℝ), in addition to a fixed conformal factor of a simple functional form; (b) the bulk path integral reduces to a path integral over Diff/SL(2, ℝ) with a Schwarzian action; (c) the low temperature free energy qualitatively agrees with that of the SYK model. We show, up to quadratic order, how to couple an infinite series of bulk scalars to the Polyakov model and show that it reproduces the coupling of the higher modes of the SYK model with the NG bosons.

Lagrangian averaging with geodesic mean

This paper revisits the derivation of the Lagrangian averaged Euler (LAE), or Euler-α equations in the light of an intrinsic definition of the averaged flow map as the geodesic mean on the volume-preserving diffeomorphism group. Under the additional assumption that first-order fluctuations are statistically isotropic and transported by the mean flow as a vector field, averaging of the kinetic energy Lagrangian of an ideal fluid yields the LAE Lagrangian. The derivation presented here assumes a Euclidean spatial domain without boundaries.

Current algebra, statistical mechanics and quantum models

Results obtained in the past for free boson systems at zero and nonzero temperatures are revisited to clarify the physical meaning of current algebra reducible functionals which are associated to systems with density fluctuations, leading to observable effects on phase transitions.To use current algebra as a tool for the formulation of quantum statistical mechanics amounts to the construction of unitary representations of diffeomorphism groups. Two mathematical equivalent procedures exist for this purpose. One searches for quasi-invariant measures on configuration spaces, the other for a cyclic vector in Hilbert space. Here, one argues that the second approach is closer to the physical intuition when modelling complex systems. An example of application of the current algebra methodology to the pairing phenomenon in two-dimensional fermion systems is discussed.

Simple length rigidity for Kleinian surface groups and applications

We prove that a Kleinian surface group is determined, up to conjugacy in the isometry group of \mathbb H^3 , by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental group of a compact, acylindrical, hyperbolizable 3-manifold M is similarly determined by the translation lengths of images of elements of \pi_1(M) represented by simple curves on the boundary of M . As a second application, we show the group of diffeomorphisms of quasifuchsian space which preserve the renormalized pressure intersection is generated by the (extended) mapping class group and complex conjugation.

Braid groups and discrete diffeomorphisms of the punctured disk

We show that the group cohomology of the diffeomorphisms of the disk with $n$ punctures has the cohomology of the braid group of $n$ strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}_{2g+2}\hookrightarrow \mathrm{Mod}_{g,2}$ to a group homomorphism between diffeomorphism groups.

Infinite loop spaces from operads with homological stability

Motivated by the operad built from moduli spaces of Riemann surfaces, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions. We prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces.The recent, strong homological stability results of Galatius and Randal-Williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability. As a consequence diffeomorphism groups and mapping class groups are shown to give rise to infinite loop spaces. Furthermore, the map to K-theory defined by the action of the diffeomorphisms on the middle dimensional homology is shown to be a map of infinite loop spaces.

Homological stability and stable moduli of flat manifold bundles

We prove that the group homology of the diffeomorphism group of #gSn×Sn\\int(D2n) as a discrete group is independent of g in a range, provided that n>2. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford–Morita–Miller classes lift to a secondary R/Z-invariants for flat (#gSn×Sn)-bundles provided g≫0.

The Lp–diameter of the group of area-preserving diffeomorphisms of S2

We show that for each $p \geq 1,$ the $L^p$-metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of Shnirelman from 1985. Our methods extend to yield stronger results on the large-scale geometry of the corresponding metric space, completing an answer to a question of Kapovich from 2012. Our proof uses configuration spaces of points on the two-sphere, quasi-morphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map $X_4(\mathbb{C} P^1) \to \mathcal{M}_{0,4} \cong \mathbb{C} P^1 \setminus \{\infty,0,1\}$ from the configuration space of $4$ points on $\mathbb{C} P^1$ to the moduli space of complex rational curves with $4$ marked points.

Stable homology of surface diffeomorphism groups made discrete

We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that $C^{\infty}$-diffeomorphisms and volume preserving diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology of $C^{\infty}$-diffeomorphims of surfaces as discrete groups is the same as homology of certain infinite loop space related to Haefliger's classifying space of foliations of codimension 2. We use this infinite loop space to obtain new results about (non)triviality of characteristic classes of flat surface bundles.

Calabi quasimorphisms for monotone coadjoint orbits

We show the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group with Kostant–Kirillov–Souriau form. We show that this result follows from positivity results of Gromov–Witten invariants and the fact that the quantum product of Schubert classes is never zero.

Finite orbits for nilpotent actions on the torus

A homeomorphism of the $2$-torus with Lefschetz number different from zero has a fixed point. We give a version of this result for nilpotent groups of diffeomorphisms. We prove that a nilpotent group of $2$-torus diffeomorphims has finite orbits when the group has some element with Lefschetz number different from zero.

Chern-Ricci invariance along G-geodesics

Over a compact oriented manifold, the space of Riemannian metrics and normalised positive volume forms admits a natural pseudo-Riemannian metric $G$, which is useful for the study of Perelman's $\mathcal{W}$ functional. We show that if the initial speed of a $G$-geodesic is $G$-orthogonal to the tangent space to the orbit of the initial point, under the action of the diffeomorphism group, then this property is preserved along all points of the $G$-geodesic. We show also that this property implies preservation of the Chern-Ricci form along such $G$-geodesics, under the extra assumption of complex aniti-invariant initial metric variation and vanishing of the Nijenhuis tensor along the $G$-geodesic. This result is useful for a slice type theorem needed for the proof of the dynamical stability of the Soliton-Kahler-Ricci flow.

Finite subgroups of Ham and Symp

Let $$(X,\omega )$$ be a compact symplectic manifold of dimension 2n and let $${\text {Ham}}(X,\omega )$$ be its group of Hamiltonian diffeomorphisms. We prove the existence of a constant C, depending on X but not on $$\omega $$ , such that any finite subgroup $$G\subset {\text {Ham}}(X,\omega )$$ has an abelian subgroup $$A\subseteq G$$ satisfying $$[G:A]\le C$$ , and A can be generated by n elements or fewer. If $$b_1(X)=0$$ we prove an analogous statement for the entire group of symplectomorphisms of $$(X,\omega )$$ . If $$b_1(X)\ne 0$$ we prove the existence of a constant $$C'$$ depending only on X such that any finite subgroup $$G\subset {\text {Symp}}(X,\omega )$$ has a subgroup $$N\subseteq G$$ which is either abelian or 2-step nilpotent and which satisfies $$[G:N]\le C'$$ . These results are deduced from the classification of the finite simple groups, the topological rigidity of hamiltonian loops, and the following theorem, which we prove in this paper. Let E be a complex vector bundle over a compact, connected, smooth and oriented manifold M; suppose that the real rank of E is equal to the dimension of M, and that $$\langle e(E),[M]\rangle \ne 0$$ , where e(E) is the Euler class of E; then there exists a constant $$C''$$ such that, for any prime p and any finite p-group G acting on E by vector bundle automorphisms preserving an almost complex structure on M, there is a subgroup $$G_0\subseteq G$$ satisfying $$M^{G_0}\ne \emptyset $$ and $$[G:G_0]\le C''$$ .

Non-stationary smooth geometric structures for contracting measurable cocycles

We implement a differential-geometric approach to normal forms for contracting measurable cocycles to $\operatorname{Diff}^{q}(\mathbb{R}^{n},\mathbf{0})$, $q\geq 2$. We obtain resonance polynomial normal forms for the contracting cocycle and its centralizer, via $C^{q}$ changes of coordinates. These are interpreted as non-stationary invariant differential-geometric structures. We also consider the case of contracted foliations in a manifold, and obtain $C^{q}$ homogeneous structures on leaves for an action of the group of subresonance polynomial diffeomorphisms together with translations.

Conjugacy of Smale-Vietoris diffeomorphisms using a conjugacy of endomorphisms

Conjugacy of Smale-Vietoris diffeomorphisms using a conjugacy of endomorphisms

Diffeomorphism groups of compact convex sets

For K⊆Rn a compact convex subset with non-empty interior, let Diff∂K(K) be the group of all C∞-diffeomorphisms of K which fix ∂K pointwise. We show that Diff∂K(K) is a C0-regular infinite-dimensional Lie group. As a byproduct, we obtain results concerning solutions to ordinary differential equations on compact convex sets.

Differential-geometrical approach to the dynamics of dissipationless incompressible Hall magnetohydrodynamics: II. Geodesic formulation and Riemannian curvature analysis of hydrodynamic and magnetohydrodynamic stabilities

In this study, the dynamics of a dissipationless incompressible Hall magnetohydrodynamic (HMHD) medium are formulated as geodesics on a direct product of two volume-preserving diffeomorphism groups. Formulations are given for the geodesic and Jacobi equations based on a linear connection with physically desirable properties, which agrees with the Levi-Civita connection. Derivations of the explicit normal-mode expressions for the Riemannian metric, Levi-Civita connection, and related formulae and equations are also provided using the generalized Elsässer variables (GEVs). Examinations of the stabilities of the hydrodynamic (HD, ) and magnetohydrodynamic (MHD, ) motions and the Hall-term effect in terms of the Jacobi equation and the Riemannian sectional curvature tensor are presented, where α represents the Hall-term strength parameter. It is very interesting that the sectional curvatures of the MHD and HMHD systems between two GEV modes were found to take both the positive (stable) and negative (unstable) values, while that of the HD system between two complex helical waves was observed to be negative definite. Moreover, for the MHD case, negative sectional curvatures were found to occur only when mode interaction was ‘local’, i.e. the wavenumber moduli of the main flow (say p) and perturbation (say k) were relatively close to each other. However, in the nonlocal limit ( or ), the sectional curvatures were always positive. This result leads to the conjecture that the MHD interactions mainly excite wavy or non-growing motions; however, some local interactions cause dynamical instability that leads to chaotic or turbulent plasma motions. Additionally, it was found that the tendencies of the effects are opposite between the ion cyclotron and whistler modes. Comparison with the energy-Casimir method is also discussed using a remarkable constant of motion which relates the Riemannian curvature to the second variation of the Hamiltonian.

On stable transitivity of finitely generated groups of volume-preserving diffeomorphisms

In this paper, we provide a new criterion for the stable transitivity of volume-preserving finite generated groups on any compact Riemannian manifold. As one of our applications, we generalize a result of Dolgopyat and Krikorian [On simultaneous linearization of diffeomorphisms of the sphere.Duke Math. J. 136(2007), 475–505] and obtain stable transitivity for random rotations on the sphere in any dimension. As another application, we show that for$\infty \geq r\geq 2$, for any$C^{r}$volume-preserving partially hyperbolic diffeomorphism$g$on any compact Riemannian manifold$M$having sufficiently Hölder stable or unstable distribution, for any sufficiently large integer$K$and for any$(f_{i})_{i=1}^{K}$in a$C^{1}$open$C^{r}$dense subset of$\text{Diff}^{r}(M,m)^{K}$, the group generated by$g,f_{1},\ldots ,f_{K}$acts transitively.