Jon Wolfson

Pseudo-holomorphic maps and bubble trees

This paper proves a strong convergence theorem for sequences of pseudo-holomorphic maps from a Riemann surface to a symplectic manifoldN with tamed almost complex structure. (These are the objects used by Gromov to define his symplectic invariants.) The paper begins by developing some analytic facts about such maps, including a simple new isoperimetric inequality and a new removable singularity theorem.The main technique is a general procedure for renormalizing sequences of maps to obtain “bubbles on bubbles.” This is a significant step beyond the standard renormalization procedure of Sacks and Uhlenbeck. The renormalized maps give rise to a sequence of maps from a “bubble tree”—a map from a wedge Σ V S2 V S2 V ... →N. The main result is that the images of these renormalized maps converge in L1,2 ∪C° to the image of a limiting pseudo-holomorphic map from the bubble tree. This implies several important properties of the bubble tree. In particular, the images of consecutive bubbles in the bubble tree intersect, and if a sequence of maps represents a homology class then the limiting map represents this class.While the main focus is on holomorphic maps, the bubble tree construction applies to other conformally invariant problems, including minimal surfaces and Yang-Mills fields.

Sympletic normal connect sum

Since the publication in 1985 of Gromov’s paper [G1] on pseudo-holomorphic curves in symplectic manifolds there has been an increased interest in symplectic manifolds and symplectic topology. In particular, compact symplectic manifolds have become a focus of much study. In 1977 Thurston [T] gave an example of a compact symplectic manifold with first Betti number three, showing that not all compact symplectic manifolds admit a Kahler structure. However, the difference between the family of compact symplectic manifolds and compact Kahler manifolds remains unclear. In fact there are essentially only two general procedures for constructing compact symplectic manifolds: the symplectic fibration construction, originally due to Thurston, and blowing up along symplectic submanifolds, introduced by Gromov [G2]. Recently R. Gompf has introduced a new construction. He considers two symplectic 4-manifolds each containing the compact surface Σ symplectically embedded with trivial normal bundle. By the symplectic neighborhood theorem a tubular neighborhood of Σ in each 4-manifold is symplectomorphic to Σ×D2 equipped with the product symplectic structure. It follows then that the complements of the tubular neighborhoods of Σ in the symplectic 4-manifolds can be symplectically glued together along tubular shell neighborhoods of Σ by the map Id × φ where φ is an area preserving map of the annulus which interchanges the boundaries. Gompf proceeded by using this construction to show that a compact simply-connected 4-manifold not admitting any complex structure, which he constructed with T. Mrowka [GM], admits a symplectic structure. He thus produced the first example of a compact simply-connected symplectic 4-manifold not admitting any Kahler structure. In this paper we introduce a construction of four dimensional symplectic manifolds, that we call symplectic normal connect sum which generalizes Gompf’s construction. Our procedure constructs a new symplectic 4manifold X = X−1#ΨX1 from pairs (Xi,Σi), i = −1, 1, where the Xi are symplectic 4-manifolds and the Σi are compact embedded symplectic surfaces of genus g and of self-intersection n (for i = 1) and −n (for i = −1), n ≥ 0. We symplectically glue the complements of tubular neighborhoods of Σ−1 in X−1 and Σ1 in X1 along tubular shell neighborhoods of Σ−1 and

Symplectic gluing along hypersurfaces and resolution of isolated orbifold singularities

This paper is concerned with two themes of symplectic topology. The first is the development of techniques to construct symplectic manifolds and, in particular, compact symplectic 4-manifolds. The second is the resolution of symplectic singularities and, in particular, the resolution of isolated singularities in symplectic 4-manifolds. On the first topic we prove a theorem which allows the gluing of two symplectic manifolds along a special class of hypersurfaces that we call o-compatible hypersurfaces. Let (X, og) be a symplectic 2n-manifold and M c X a hypersurface with a fixed point free S 1-action. M is called o2compatible if the orbits of the action lie in the null directions of og[u. An ~o-compatible hypersurface M has a canonical co-orientation. Hence, if M is a separating hypersurface, then M divides X into distinguished components X and X +. In dimension 4, our main gluing theorem is as follows.

The fundamental group of manifolds of positive isotropic curvature and surface groups

In this paper we study the topology of compact manifolds of positive isotropic curvature (PIC). There are many examples of non-simply connected compact manifolds with positive isotropic curvature. We prove that the fundamental group of a compact Riemannian manifold with PIC, of dimension ≥ 5, does not contain a subgroup isomorphic to the fundamental group of a compact Riemann surface. The proof uses stable minimal surface theory.

Eigenvalue Gap Theorems for a Class of Nonsymmetric Elliptic Operators on Convex Domains

Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose first order terms have coefficients with compact support in the open domain. The eigenvalue gap is bounded below by the gap of an associated Sturm-Liouville problem on a closed interval.

FILL RADIUS AND THE FUNDAMENTAL GROUP

In this note we relate the geometric notion of fill radius with the fundamental group of the manifold. We prove: Suppose that a closed Riemannian manifold M satisfies the property that its universal cover has bounded fill radius. Then the fundamental group of M is virtually free. We explain the relevance of this theorem to some conjectures on positive isotropic curvature and 2-positive Ricci curvature.

Double points and the proper transform in symplectic geometry

Abstract In his book, Partial Differential Relations, Gromov introduced the symplectic analogue of the complex analytic operations of blowing up and blowing down. Gromov proposed, in Sect. 3.4.4(D) of Partial Differential Relations, a program for resolving the singularities of symplectic immersions with symplectic crossings via blowing up, in exact analogue with the well known complex analytic technique. The purpose of this note is to show that this program cannot work. We show that there are symplectically immersed surfaces in symplectic 4-manifolds which do not have a symplectically embedded proper transform in any blow up of the four manifold.

A Simple Proof of Frobenius Theorem

Frobenius Theorem, as stated in Y. Matsushima, Differential Manifolds, Marcel Dekker, N.Y., 1972, p. 167, is the following: Let D be an r-dimensional differential system on an n-dimensional manifold M. Then D is completely integrabte if and only if for every local basis {X1,...,Xr} of D on any open set V of M , there are C∞ -functvons c ij k on V such that we have

Minimizing Area Among Lagrangian Surfaces: The Mapping Problem

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