Michael Hutchings

Proof of the Double Bubble Conjecture

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in R 3 .

Circle-valued Morse theory and Reidemeister torsion

Let X be a closed manifold with (X )=0 , and let f : X! S 1 be a circlevalued Morse function. We dene an invariant I which counts closed orbits of the gradient of f , together with flow lines between the critical points. We show that our invariant equals a form of topological Reidemeister torsion dened by Turaev [28]. We proved a similar result in our previous paper [7], but the present paper renes this by separating closed orbits and flow lines according to their homology classes. (Previously we only considered their intersection numbers with a xed level set.) The proof here is independent of the proof in [7], and also simpler. Aside from its Morse-theoretic interest, this work is motivated by the fact that when X is three-dimensional and b1(X) > 0, the invariant I equals a counting invariant I3(X) which was conjectured in [7] to equal the Seiberg{Witten invariant of X . Our result, together with this conjecture, implies that the Seiberg{Witten invariant equals the Turaev torsion. This was conjectured by Turaev [28] and renes the theorem of Meng and Taubes [14]. AMS Classication numbers Primary: 57R70 Secondary: 53C07, 57R19, 58F09

CIRCLE-VALUED MORSE THEORY, REIDEMEISTER TORSION, AND SEIBERG—WITTEN INVARIANTS OF 3-MANIFOLDS

Abstract Let X be a closed oriented Riemannian manifold with χ(X)=0 and b1(X)>0, and let φ : X→S 1 be a circle-valued Morse function. Under some mild assumptions on φ, we prove a formula relating 1. the number of closed orbits of the gradient flow of φ in different homology classes; 2. the torsion of the Novikov complex, which counts gradient flow lines between critical points of φ; and 3. a kind of Reidemeister torsion of X determined by the homotopy class of φ. When dim(X)=3, we state a conjecture related to Taubes’s “SW=Gromov” theorem, and we use it to deduce (for closed manifolds, modulo signs) the Meng–Taubes relation between the Seiberg-Witten invariants and the “Milnor torsion” of X.

The double bubble conjecture

The classical isoperimetric inequality states that the surface of smallest area enclosing a given volume in R is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 2π/3.

The Weinstein conjecture for stable Hamiltonian structures

We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3-manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y. We prove that if Y is not a T^2-bundle over S^1, then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3-manifold with a contact form such that all Reeb orbits are nondegenerate and elliptic, then Y is a lens space. Related arguments show that if Y is a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate, and if Y is not a lens space, then there exist at least three distinct embedded Reeb orbits.

Rounding corners of polygons and the embedded contact homology of T 3

The embedded contact homology (ECH) of a 3‐manifold with a contact form is a variant of Eliashberg‐Givental‐Hofer’s symplectic field theory, which counts certain embedded J ‐holomorphic curves in the symplectization. We show that the ECH of T 3 is computed by a combinatorial chain complex which is generated by labeled convex polygons in the plane with vertices at lattice points, and whose differential involves “rounding corners”. We compute the homology of this combinatorial chain complex. The answer agrees with the Ozsvath‐Szabo Floer homology HF C .T 3 /.

The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature

We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.

The periodic Floer homology of a Dehn twist.

The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts certain embedded pseudoholomorphic curves in R cross the mapping torus. It is conjectured to recover the Seiberg-Witten Floer homology of the mapping torus for most spin-c structures, and is related to a variant of contact homology. In this paper we compute the periodic Floer homology of some Dehn twists. AMS Classification 57R58; 53D40, 57R50

The structure of area-minimizing double bubbles

We show that the least area required to enclose two volumes in ℝn orSn forn ≥ 3 is a strictly concave function of the two volumes. We deduce that minimal double bubbles in ℝn have no empty chambers, and we show that the enclosed regions are connected in some cases. We give consequences for the structure of minimal double bubbles in ℝn. We also prove a general symmetry theorem for minimal enclosures ofm volumes in ℝn, based on an idea due to Brian White.

Taubes’s proof of the Weinstein conjecture in dimension three

Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? No, according to counterexamples by K. Kuperberg and others. On the other hand there is a special class of vector fields, called Reeb vector fields, which are associated to contact forms. The three-dimensional case of the Weinstein conjecture asserts that every Reeb vector field on a closed oriented three-manifold has a closed orbit. This conjecture was recently proved by Taubes using Seiberg-Witten theory. We give an introduction to the Weinstein conjecture, the main ideas in Taubes’s proof, and the bigger picture into which it fits. Taubes’s proof of the Weinstein conjecture is the culmination of a large body of work, both by Taubes and by others. In an attempt to make this story accessible to nonspecialists, much of the present article is devoted to background and context, and Taubes’s proof itself is only partially explained. Hopefully this article will help prepare the reader to learn the full story from Taubes’s paper [62]. More exposition of this subject (which was invaluable in the preparation of this article) can be found in the online video archive from the June 2008 MSRI hot topics workshop [44], and in the article by Auroux [5]. Below, in §1–§3 we introduce the statement of the Weinstein conjecture and discuss some examples. In §4–§6 we discuss a natural strategy for approaching the Weinstein conjecture, which proves it in many but not all cases, and provides background for Taubes’s work. In §7 we give an overview of the big picture surrounding Taubes’s proof of the Weinstein conjecture. Readers who already have some familiarity with the Weinstein conjecture may wish to start here. In §8–§9 we recall necessary material from Seiberg-Witten theory. In §10 we give an outline of Taubes’s proof, and in §11 we explain some more details of it. To conclude, in §12 we discuss some further results and open problems related to the Weinstein conjecture. 1. Statement of the Weinstein conjecture The Weinstein conjecture asserts that certain vector fields must have closed orbits. Before stating the conjecture at the end of this section, we first outline its origins. This is discussion is only semi-historical, because only a sample of the relevant works will be cited, and not always in chronological order. 1.1. Closed orbits of vector fields. Let Y be a closed manifold (in this article all manifolds and all objects defined on them are smooth unless otherwise stated), 2000 Mathematics Subject Classification. 57R17,57R57,53D40. Partially supported by NSF grant DMS-0806037.