Non-isotopic symplectic surfaces in product 4-manifolds

Let be a closed Riemann surface of genus . Generalizing Ivan Smith's construction, we give the first examples of an infinite family of homotopic but pairwise non-isotopic symplectic surfaces of even genera inside the product symplectic -manifolds , where and .

     

The geography of spin symplectic 4-manifolds

In this paper we construct a family of simply connected spin non-complex symplectic 4-manifolds which cover all but finitely many allowed lattice points () lying in the region . Furthermore, as a corollary, we prove that there exist infinitely many exotic smooth structures on for all n large enough. Received: 29 August 2000 / in final form: 15 August 2001 / Published online: 28 February 2002     

Smoothly embedded spheres in symplectic 4-manifolds

We characterize rational or ruled surfaces among all symplectic 4-manifolds by the existence of certain smoothly embedded spheres.

     

Fomenko-Zieschang invariants of integrable systems with symplectic singularities

The paper is devoted to the study of integrable Hamiltonian systems which arise on symplectic manifolds with degenerate singularities. The results obtained extend the subject domain of the Fomenko-Zieschang theory of invariants.     

Filling by holomorphic curves in symplectic 4-manifolds

We develop a general framework for embedded (immersed) -holomorphic curves and a systematic treatment of the theory of filling by holomorphic curves in 4-dimensional symplectic manifolds. In particular, a deformation theory and an intersection theory for -holomorphic curves with boundary are developed. Bishop's local filling theorem is extended to almost complex manifolds. Existence and uniqueness of global fillings are given complete proofs. Then they are extended to the situation with nontrivial -holomorphic spheres, culminating in the construction of singular fillings.

     

Ruled 4-manifolds and isotopies of symplectic surfaces

We study symplectic surfaces in ruled symplectic 4-manifolds which are disjoint from a given symplectic section. As a consequence, in any symplectic 4-manifold, two homologous symplectic surfaces which are C ⁰ close must be Hamiltonian isotopic.     

Exotic involutions on symplectic 4-manifolds and G-monopole invariants

Using the G-monopole invariant, we shall show that any anti-holomorphic involution on a closed symplectic 4-manifold is not diffeomorphic to any holomorphic involution. As a corollary, we shall give a way to construct exotic smooth structures.     

Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.     

Geography of simply connected spin symplectic 4-manifolds,II

Building upon our early work, we construct infinitely many new smooth structures on closed simply connected spin 4-manifolds with nonnegative signature.     

Symplectic invariants for curves and integrable systems in similarity symplectic geometry

In this paper, similarity symplectic geometry for curves is proposed and studied. Explicit expressions of the symplectic invariants, Frenet frame and Frenet formulae for curves in similarity symplectic geometry are obtained by using the equivariant moving frame method. The relationships between the Euclidean symplectic invariants, Frenet frame, Frenet formulae and the similarity symplectic invariants, Frenet frame, Frenet formulae for curves are established. Invariant curve flows in four-dimensional similarity symplectic geometry are also studied. It is shown that certain intrinsic invariant curve flows in four-dimensional similarity symplectic geometry are related to the integrable Burgers and matrix Burgers equations.     

On the geography of symplectic 6-manifolds

The article investigates the geography of closed, connected and simply connected, six-dimensional manifolds. It is proved that any triple of integers satisfying some necessary arithmetical restrictions occurs as the Chern triple of such a manifold. The main tools used for producing the examples are the symplectic connected sum and the symplectic blow-up. Received: 28 May 1998 / Revised version: 22 January 1999     

Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry

We first build the moduli spaces of real rational pseudo-holomorphic curves in a given real symplectic 4-manifold. Then, following the approach of Gromov and Witten [3, 19, 11], we define invariants under deformation of real symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of points. Mathematics Subject Classification (2000) 14N10, 14P25, 53D05, 53D45     

Constructing the Kähler and the symplectic structures from certain spinors on 4-manifolds

We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spin structure implies that the underlying smooth manifold admits a Kähler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that the structure in which the non-zero parallel spinor lives is equivalent to the canonical spin structure associated to the Kähler structure.

     

Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry

Following the approach of Gromov and Witten, we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves in a given homology class passing through a given real configuration of points. To cite this article: J.-Y. Welschinger, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Remarks on integrable systems

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ? n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.     

Twistors, 4-symmetric spaces and integrable systems

An order four automorphism of a Lie algebra gives rise to an integrable system introduced by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space. As applications, we find that surfaces with holomorphic mean curvature in 4-dimensional real or complex space forms constitute an integrable system as do Hamiltonian stationary Lagrangian surfaces in 4-dimensional Hermitian symmetric spaces (this last providing a conceptual explanation of a result of Hélein-Romon).     

Q-curvature flow on 4-manifolds

We formulate an appropriate gradient flow in order to study the evolution of the Q-curvature to a prescribed function on a 4-manifold. For a class of prescribed functions, we show convergence and describe the asymptotic behaviour at infinity.