Joel Hass

Least area incompressible surfaces in 3-manifolds

Let M be a Riemannian manifold and let F be a closed surface. A map f: F---,M is called least area if the area of f is less than the area of any homotopic map from F to M. Note that least area maps are always minimal surfaces, but that in general minimal surfaces are not least area as they represent only local stationary points for the area function. The existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group [18, 17]. In this paper we shall consider the possible singularities of such immersions. Our results show that the general philosophy is that least area surfaces intersect least, meaning that the intersections and self-intersections of least area immersions are as small as their homotopy classes allow, when measured correctly. One should note that evidence supporting this view had been found by Meeks-Yau in their embedding theorems for minimal disks and 2-spheres [13, 143 . Our main result asserts that if a least area immersion is homotopic to an embedding, then it has no self-intersections, which clearly exemplifies the above philosophy. The precise result is the following.

SHORTENING CURVES ON SURFACES

METHODS of shortening a curve in a manifold have been used to establish the existence of closed geodesics, and in particular of simple closed geodesics on 2-spheres. For this purpose, a curve evolution process should (a) not increase the number of self-intersections of a curve, (b) exist for all time or until a curve collapses to a point, (c) shorten curves sufficiently fast so that curves which exist for all time converge to a geodesic, and (d) depend continuously on the choice of initial curve. Birkhoff originated what is now known as the Birkhoff curve shortening process, where midpoints of polygonal approximations to a curve are successively connected by geodesic segments [4]. This type of shortening has the advantage that (b), (c) and (d) are easy to establish, but the disadvantage that (a) seems difficult to arrange. A process of evolving a curve on a surface by its curvature is perhaps the most natural flow. Short term existence is easy to establish for this flow, but long term existence involves deep questions in PDEs and geometry. This flow has recently been studied with considerable success in a series of papers [7,8,9, 11. All four of the desired properties have been shown to hold for the flow by curvature of an embedded curve on a Riemannian surface. For non-embedded curves in Riemannian surfaces, some open questions remain about the types of singularities which may develop in the curvature flow. In particular, it is not known whether arcs of double points can be created. In this paper we introduce a new curve shortening flow. Like the Birkhoff process, this flow involves replacing arcs of a curve with geodesic segments. Unlike the Birkhoff process, it picks out its piecewise-geodesic structure purely from the geometry of the image manifold rather than from a parametrization of the curve. This flow, which we call the disk Jlow, is developed in

The number of Reidemeister moves needed for unknotting

1. In

The computational complexity of knot genus and spanning area

2 we use the flow to solve a purely topological problem concerning intersections of curves on surfaces. Turaev [17] has posed the problem in the following form:

Intersections of curves on surfaces

There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embedded in the 1-skeleton of the interior of a compact, orientable, triangulated P L 3-manifold M. There is a positive constant c2 such that for each t � 1, if M consists of t tetrahedra, and K is unknotted, then there is a sequence of at most 2 c 2t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M. We obtain explicit values for c1 and c2.

The existence of least area surfaces in 3-manifolds

We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show that deciding whether a curve in a metrized PL 3-manifold bounds a surface of area less than a given constant C is NP-hard.

The double bubble conjecture

AbstractThe authors consider curves on surfaces which have more intersections than the least possible in their homotopy class. Theorem 1.Let f be a general position arc or loop on an orientable surface F which is homotopic to an embedding but not embedded. Then there is an embedded 1-gon or 2-gon on F bounded by part of the image of f. Theorem 2.Let f be a general position arc or loop on an orientable surface F which has excess self-intersection. Then there is a singular 1-gon or 2-gon on F bounded by part of the image of f.Examples are given showing that analogous results for the case of two curves on a surface do not hold except in the well-known special case when each curve is simple.

Immersions of surfaces in 3-manifolds

This paper presents a new and unified approach to the existence theorems for least area surfaces in 3-manifolds. Introduction. A surface F smoothly embedded or immersed in a Riemannian manifold M is minimal if it has mean curvature zero at all points. It is a least area surface in a class of surfaces if it has finite area which realizes the infimum of all possible areas for surfaces in this class. The connection between these two ideas is that a surface which is of least area in a reasonable class of surfaces must be minimal. The converse is false; minimal surfaces are in general only critical points for the area function. There are close analogies between these two concepts and the theory of geodesics in a Riemannian manifold. Minimal surfaces correspond to geodesics, and least area surfaces correspond to geodesic arcs or closed geodesics which have shortest length in some class of paths. Any geodesic A in a Riemannian manifold M has the property that it is locally shortest, i.e., if P and Q are nearby points on A, then the subarc of A which joins P and Q is the shortest path in M from P to Q. It can also be proved that minimal surfaces are locally of least area, but the proof is difficult and involves substantial knowledge of the theory of partial differential equations. There are now a large number of theorems asserting the existence of surfaces of least area in various classes. Surfaces of this type have become an important tool in 3-dimensional topology. In this paper we present a new approach to the proofs of these existence theorems. This yields a simplified and unified method for the proof of the existence of minimal surfaces in 3-dimensional Riemannian manifolds. In 1930 Douglas [Do] and Rado [Ra] independently showed that a simple closed curve in RI which bounds a disk of finite area bounds a disk of least possible area. This result was extended by Morrey [MoI] in 1948 to a general class of Riemannian manifolds, the homogeneously regular manifolds, a class which includes all closed manifolds. Work of Osserman [0] and Gulliver [G] later showed that least area disks in 3-manifolds were immersed in their interiors. More recently there has been a series of new existence results for closed surfaces of least area in manifolds of any dimension. It follows from work of Sacks and Uhlenbeck [S-UI] that if M is closed and 7r2(M) is nonzero then there is an essential map of the 2-sphere into M which has least area among all essential maps. Sacks and Uhlenbeck [S-UII] and Schoen and Yau [S-Y] independently showed that if f: F -* M is a map of a closed orientable surface F, not the 2-sphere, into a closed Riemannian manifold M, such Received by the editors June 23, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 53A10; Secondary 57M35. The first author was supported by NSF Postdoctoral Fellowship DMS84-14097. ?)1988 American Mathematical Society 0002-9947/88

Double bubbles minimize

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Linear perturbation methods for topologically consistent representations of free-form surface intersections

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