On properly essential classical conformal diffeomorphism groups

In this article, we prove that various classical conformal diffeomorphism groups, which are known to be essential (Banyaga, J Geom 68(1–2):10–15, 2000), are in fact properly essential. This is a consequence of a local criterion on a conformal diffeomorphism in the form of a cohomological equation. Furthermore, we study the orbit of a tensor field under the action of the conformal diffeomorphism group for these classical conformal structures. On every closed contact manifold, we find conformal contact forms that are not diffeomorphic.     

An analytic version of the De Rham theorem

LetV be a real analytic paracompact variety; in §1 of this paper we prove that: $$H^q (V,R) \approx \frac{{closed analytic differentiable q - forms on V}}{{exact analytic differentiable q - forms on V}}$$ Then we prove that the closed (exact) analytic differentiableq-forms onV are dense, in the Whitney topology, in the set of closed (exact) differentiableq-forms onV. We also consider the problem of extending closed (exact) analytic differentiableq-forms, defined on a subvarietyX ofV, to closed (exact) analytic forms defined onV.     

Commutators of contactomorphisms

The group of volume preserving diffeomorphisms, the group of symplectomorphisms and the group of contactomorphisms constitute the classical groups of diffeomorphisms. The first homology groups of the compactly supported identity components of the first two groups have been computed by Thurston and Banyaga, respectively. In this paper we solve the long standing problem on the algebraic structure of the third classical diffeomorphism group, i.e. the contactomorphism group. Namely we show that the compactly supported identity component of the group of contactomorphisms is perfect and simple (if the underlying manifold is connected). The result could be applied in various ways.     

Geodesic completeness for Sobolev H s -metrics on the diffeomorphism group of the circle

We prove that the weak Riemannian metric induced by the fractional Sobolev norm H ^s on the diffeomorphism group of the circle is geodesically complete, provided that s > 3/2.     

Hamiltonian dynamics on convex symplectic manifolds

We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo, Schwarz and Oh. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many non-trivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed trajectories of a charge in a magnetic field on almost all small energy levels. We shall also obtain some new Lagrangian intersection results. Partially supported by the Swiss National Foundation. Supported by the Swiss National Foundation and the von Roll Research Foundation.     

Induced Hamiltonian maps on the symplectic quotient

Let (M,ω) be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M, with no non-constant closed trajectory in time less than 1 and time-1 map fH. If the map fH descends to the symplectic quotient to a map Φ(fH) and the symplectic manifold M is exact and Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH) is bounded above by the Hofer norm of fH.     

Hamiltonian flows on the space of star-shaped curves

We provide a geometric interpretation of the KdV equation as an evolution equation on the space of closed curves in the centroaffine plane. There is a natural symplectic structure on this space and the KdV-flow is generated by a Hamiltonian given by the total centroaffine curvature. In this way we obtain another example for a soliton equation coming naturally from a differential geometric problem [1]. Furthermore, we present a group action of the diffeomorphism group of the circle on the space of closed centroaffine curves.     

Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

We study the following rigidity problem in symplectic geometry: can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser (Acta. Math. 141(1–2):17–34, 1978). Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the ‘virtually contact’ setting. By means of an Abbondandolo–Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.     

Cup-length estimate for Lagrangian intersections

In this paper, we consider the Arnold conjecture on the Lagrangian intersections of some closed Lagrangian submanifold of a closed symplectic manifold with its image of a Hamiltonian diffeomorphism. We prove that if the Hofer's symplectic energy of the Hamiltonian diffeomorphism is less than a topology number defined by the Lagrangian submanifold, then the Arnold conjecture is true in the degenerated (nontransversal) case.     

On closed manifolds admitting an Anosov diffeomorphism but no expanding map

A few years ago, the first example of a closed manifold admitting an Anosov diffeomorphism but no expanding map was given. Unfortunately, this example is not explicit and is high-dimensional, although its exact dimension is unknown due to the type of construction. In this paper, we present a family of concrete 12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map, where a nilmanifold is defined as the quotient of a 1-connected nilpotent Lie group by a cocompact lattice. We show that this family has the smallest possible dimension in the class of infra-nilmanifolds, which is conjectured to be the only type of manifolds admitting Anosov diffeomorphisms up to homeomorphism. The proof shows how to construct positive gradings from the eigenvalues of the Anosov diffeomorphism under some additional assumptions related to the rank, using the action of the Galois group on these algebraic units.     

An example of convex Hamiltonian diffeomorphism where asymptotic distance from identity is strictly greater than the minimal action

We construct a convex Hamiltonian diffeomorphism on the unit ball of cotangent bundle of Tn (n?2), where the asymptotic distance from identity is strictly greater than the minimal action.     

Cohomology of regular differential forms for affine curves

Let C be a complex affine reduced curve, and denote by H¹(C) its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant μ^′(C,x) that measures the complexity of the singularity of C at the point x, and we establish the following formula:

     

Growth of maps, distortion in groups and symplectic geometry

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism:?• The uniform norm of the differential of its n-th iteration;?• The word length of its n-th iteration, where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms.?We find lower bounds for the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology. As an application, we prove non-existence of certain non-linear symplectic representations for finitely generated groups. Oblatum 6-XII-2001 & 19-VI-2002?Published online: 5 September 2002 RID="*" ID="*"Supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

Reduced Delzant spaces and a convexity theorem

The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian G-manifold is a stratified symplectic space. Suppose 1→A→G→T→1 is an exact sequence of compact Lie groups and T is a torus. Then the reduction of a Hamiltonian G-manifold with respect to A yields a Hamiltonian T-space. We show that if the A-moment map is proper, then the convexity theorem holds for such a Hamiltonian T-space, even when it is singular. We also prove that if, furthermore, the T-space has dimension and T acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case. This paper is a concise version of a companion paper [B. Lian. B. Song, A convexity theorem and reduced Delzant spaces, math.DG/0509429].     

Maximal Linear Subspaces of Strong Self-Dual 2-forms and the Bonan 4-form

The notion of self-duality of 2-forms in 4-dimensions plays an eminent role in many areas of mathematics and physics, but although the 2-forms have a genuine meaning related to curvature and gauge-field-strength in higher dimensions also, their “self-duality” is something which is almost avoided above 4-dimensions. We show that self-duality of 2-forms is a very natural notion in higher (even) dimensions also and we prove the equivalence of some scattered and rarely used definitions in the literature. We demonstrate the usefulness of this higher self-duality by studying it in 8-dimensions and we derive a natural expression for the Bonan form in terms of self-dual 2-forms and we give an explicit expression of the local action of SO(8) on the Bonan form.     

A growth gap for diffeomorphisms of the interval

Given an orientation-preserving diffeomorphism of the interval [0;1], consider the uniform norm of the differential of itsn-th iteration. We get a function ofn called the growth sequence. Its asymptotic behaviour is an interesting invariant, which naturally appears both in geometry of the diffeomorphism groups and in smooth dynamics. Our main result is the following Gap Theorem: the growth rate of this sequence is either exponential or at most quadratic withn. Further, we construct diffeomorphisms whose growth sequence has quite irregular behaviour. This construction easily extends to arbitrary manifolds.     

Integrability of Invariant Metrics on the Diffeomorphism Group of the Circle

Each H^k Sobolev inner product (k ≥ 0) defines a Hamiltonian vector field X_k on the regular dual of the Lie algebra of the diffeomorphism group of the circle. We show that only X₀ and X₁ are bi-Hamiltonian relative to a modified Lie-Poisson structure.     

Well-posedness of modified Camassa-Holm equations

The Camassa-Holm equation can be viewed as the geodesic equation on some diffeomorphism group with respect to the invariant H¹ metric. We derive the geodesic equations on that group with respect to the invariant Hk metric, which we call the modified Camassa-Holm equation, and then study the well-posedness and dynamics of a modified Camassa-Holm equation on the unit circle S, which has some significant difference from that of Camassa-Holm equation, e.g., it does not admit finite time blowup solutions.     

An iterative algorithm for the least Frobenius norm Hermitian and generalized skew Hamiltonian solutions of the generalized coupled Sylvester-conjugate matrix equations

The iterative method of the generalized coupled Sylvester-conjugate matrix equations \(\sum\limits _{j=1}^{l}\left (A_{ij}X_{j}B_{ij}+C_{ij}\overline {X}_{j}D_{ij}\right )=E_{i} (i=1,2,\cdots ,s)\) over Hermitian and generalized skew Hamiltonian solution is presented. When these systems of matrix equations are consistent, for arbitrary initial Hermitian and generalized skew Hamiltonian matrices X _j (1), j = 1,2,? , l, the exact solutions can be obtained by iterative algorithm within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm Hermitian and generalized skew Hamiltonian solution of the problem. Finally, numerical examples are presented to demonstrate the proposed algorithm is efficient.