Isotopy invariants for smooth tori in 4-manifolds

In this paper we define invariants under smooth isotopy for certain two-dimensional knots using some refined Cerf theory. One of the invariants is the knot type of some classical knot generalizing the string number of closed braids. The other invariant is a generalization of the unique invariant of degree 1 for classical knots in 3-manifolds. Possibly, these invariants can be used to distinguish smooth embeddings of tori in some 4-manifolds but which are equivalent as topological embeddings.     

Gromov's K-area and symplectic rigidity

Supported by the United States-Israel Binational Science Foundation grant 94-00302     

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     

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.     

Circle-sum and minimal genus surfaces in ruled 4-manifolds

We describe a circle-sum construction of smoothly embedded surface in a smooth 4-manifold. We apply this construction to give a simpler solution of the minimal genus problem for nontrivial bundles over surfaces. We also treat the case of blow-ups.

     

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 .

     

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.     

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.

     

Semitoric integrable systems on symplectic 4-manifolds

Let (M,ω) be a symplectic 4-manifold. A semitoric integrable system on (M,ω) is a pair of smooth functions J,H∈C ^∞(M,ℝ) for which J generates a Hamiltonian S ¹-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce. A. Pelayo was partially supported by an NSF Postdoctoral Fellowship.     

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.

     

Curvature and injectivity radius estimates for Einstein 4-manifolds

Let denote an Einstein -manifold with Einstein constant, , normalized to satisfy . For , a metric ball, we prove a uniform estimate for the pointwise norm of the curvature tensor on , under the assumption that the -norm of the curvature on is less than a small positive constant, which is independent of , and which in particular, does not depend on a lower bound on the volume of . In case , we prove a lower injectivity radius bound analogous to that which occurs in the theorem of Margulis, for compact manifolds with negative sectional curvature, . These estimates provide key tools in the study of singularity formation for -dimensional Einstein metrics. As one application among others, we give a natural compactification of the moduli space of Einstein metrics with negative Einstein constant on a given .

     

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     

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.     

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     

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).     

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.

     

Coverings of 3-manifolds by open balls and two open solid tori

Hempel and McMillan showed that a closed 3-manifold that can be covered by three open balls is a connected sum of S³- and S²-bundles over S¹. In this paper we obtain a classification of all closed 3-manifolds that can be covered by two open balls and one open solid torus or by one open ball and two open solid tori.