William H. Meeks

The strong halfspace theorem for minimal surfaces

D. Hoffman and W.H. Meeks, III Department of Mathematics, University of Massachusetts, Amherst, MA 01003, USA In [6] Jorge and Xavier constructed examples of complete nonplanar minimal surfaces contained between two parallel planes in •3. Recently Rosenberg and Toubiana have found complete minimally immersed annuli that are proper in an open slab 1-12]. On the basis of these and other results, the question has arisen as to whether or not there could exist a properly immersed minimal surface in ~3 that was contained between two parallel planes. The answer is negative, an immediate consequence of the following theorem.

The topology of complete minimal surfaces of finite total Gaussian curvature

RECENTLY GREAT progress has been made in the classical theory of minimal surfaces in R3. For example, the proof of the embedding of the solution to Plateau’s problem for an extremal Jordan curve [lo] and the embedding of a solution to the free boundary value problem [ll], the proof of the bridge theorem [12], the regularity and finiteness of least area oriented surfaces bounding a smooth Jordan curve [5], the uniqueness and topological uniqueness of certain minimal surfaces 18, 9, 121, the proof that a complete stable minimal surface is a plane [l, 31, the existence of a collection of 4 Jordan curves which bound a continuous family of compact minimal surfaces [141, the existence of a complete minimal surface contained between two parallel planes [13] and a theorem [18] which states that if the Gauss map of a complete minimal surface misses 7 points on the ‘sphere, then the minimal surface is a plane. The aim of this work is to begin the exploration of the topological properties of complete minimal surfaces of finite total curvature. First we recall the fundamental classical result of Chern-Osserman [6] that states that such surfaces M are conformally equivalent to compact Riemann surfaces iii punctured in a finite number of points. Furthermore, they prove that the Gauss map on the surface M extends conformally to a. In particular, the theorem of Chern-Osserman implies that such surfaces have finite topological type with a finite number of topological ends and that the normal vectors at infinity on these ends are well defined. (Finite topological type means in this case that M is diffeomorphic to the interior of a compact surface with boundary.) Our first theorem gives a nice description of the topological placement in IX3 of a complete surface of finite topological type which has well defined normal vectors at infinity. We prove that the ends of such surfaces behave like the ends of the catenoid at infinity. More precisely, Theorem 1 shows that if f: M + R3 is an immersion of such a surface, then f is proper and the image f(M) viewed from infinity looks like a finite collection of flat planes (with multiplicity) that pass through the origin. For example, the catenoid viewed from infinity looks like two oppositely oriented copies of a plane passing through the origin. Theorem 1 has a natural generalization to submanifolds of arbitrary codimension in R”. This generalization which is Theorem 2 is discussed in

THE CLASSICAL PLATEAU PROBLEM AND THE TOPOLOGY OF THREE-DIMENSIONAL MANIFOLDS

LET y be a rectifiable Jordan curve in three-dimensional euclidean space. Answering an old question, whether y can bound a surface with minimal area, Douglas [l I] and Rad6[45] (independently) found a minimal surface spanning y which is parametrized by the disk. This minimal surface has minimal area among all Lipschitz maps from the disk into R3 which span y. The question whether this solution has branch points or not was finally settled by Osserman[42], who proved that there are no interior “true” branch points, and by Gulliver[lS], who proved that there are no interior “false” branch points. In 1948, Morrey[35] devised a new method to solve the Plateau problem for a map from the disk into a “homogeneously regular*’ Riemannian manifold. Moreover, he proved the interior regularity of the map in case the ambient manifold is regular, and that the map is real analytic if the ambient manifold is real analytic. The arguments of Osserman and Gulliver in addition show that Morrey’s solution has no interior branch point when the ambient manifold is three-dimensional. In 1951, Lewy[29] showed that if the Jordan curve y is also real analytic, in a real analytic manifold, then any minimal surface with boundary y is real analytic up to the boundary. Hence in this case Morrey’s map is real analytic map on the closed disk. (For a proof of Lewy’s Theorem in a general real analytic manifold, see HiIdebrandt[23].) In 1969, Hildebrandt[23] proved that the Douglas solution is smooth up to the boundary if the Jordan curve is smooth and regular. (Further improvements are due to Kinderlehrer [25], Nitsche [40], and Warschawski[55].) In [22], Heinz and Hildebrandt extended Hildebrandt’s result to minimal surfaces in general Riemannian manifolds. Once we have boundary regularity, it makes sense to ask whether Douglas’ or Morrey’s solution of the Plateau problem has a boundary branch point or not, To date, this problem has not been settled. The first partial result in this direction is due to Nitsche[40], who showed that there are only a finite number of boundary branch points for minimal surfaces with smooth boundary in R3. This was then generalized by Heinz and Hildebrandt to smooth manifolds. Gulliver and Lesley[l6] have also observed that the Douglas-Morrey solution for a real analytic curve in a real analytic manifold has no boundary branch point, using the previously mentioned result of Lewy. Despite all these results, an interesting topological question remained unsolved, namely, under what conditions is the Douglas solution an embedded surface? It has been generally conjectured that when the Jordan curve is extremal, i.e. lies on the

Complete embedded minimal surfaces of finite total curvature

We will survey what is known about minimal surfaces S ⊂ R 3, which are complete, embedded, and have finite total curvature: \(\int_s {|K|} dA < \infty \). The only classically known examples of such surfaces were the plane and the catenoid. The discovery by Costa [14, 15] early in the last decade, of a new example that proved to be embedded sparked a great deal of research in this area. Many new examples have been found, even families of them, as will be described below. The central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples.

Finite group actions on 3-manifolds

If G is a finite group acting smoothly on a closed surface F, it is well known that G leaves invariant some Riemannian metric of constant curvature on F. Thus any action of G on the 2-sphere S 2 is conjugate in Diff(S 2) to an orthogonal action. If G acts on the torus SX• S 1, there is a G-invariant flat metric on S a • S 1, and if G acts on a surface F with negative Euler number, then F admits a G-invariant hyperbolic metric. Recently Thurston, [Th 1, Th 2, Th 3], has described the eight 3-dimensional geometries which provide geometric structures for closed 3-manifolds in the same way that the 2-sphere S 2, the Euclidean plane E 2 and the hyperbolic plane H 2 provide geometric structures for surfaces. See also the survey article by Scott [Sc4]. Thurston also conjectured that if M is a closed 3-manifold with a geometric structure modelled on one of these eight geometries, say X, then any smooth action of a finite group G on M should leave invariant some metric on M inducing the geometry X. We will say that G preserves the geometric structure on M in this case. It should be noted that the restriction to smooth actions of G on M is essential. For Bing [Bi] showed that there are involutions of S 3 whose fixed set is a wild 2-sphere. However, in dimension two, it was proved by Eilenberg [Ei] that any action of a finite group on a surface is conjugate to a smooth action. In this paper, our main result asserts that Thurstons conjecture holds for five of the eight geometries. The result is the following.

Embedded minimal surfaces of finite topology

In this paper we prove that any complete, embedded minimal surface M in R 3 with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface M with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion M . In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior ofM .

CONSTANT MEAN-CURVATURE SURFACES IN HYPERBOLIC SPACE

Supported by the National Science Foundation grant DMS-8808002. Supported by the National Science Foundation grant DMS-8908064. The research described in this paper was supported by research grant DE-FG0286ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS-8900285. Supported by the National Science Foundation grant DMS-8800414.

The minimal lamination closure theorem

We prove that the closure of a complete embedded minimal surface M in a Riemannian three-manifold N has the structure of a minimal lamination, when M has positive injectivity radius. When N is R, we prove that such a surface M is properly embedded. Since a complete embedded minimal surface of finite topology in R has positive injectivity radius, the previous theorem implies a recent theorem of Colding and Minicozzi: A complete embedded minimal surface of finite topology in R is proper. More generally, we prove that if M is a complete embedded minimal surface of finite topology and N has nonpositive sectional curvature (or is the Riemannian product of a homogeneously regular Riemannian surface with R), then the closure of M has the structure of a minimal lamination. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42

EMBEDDED MINIMAL-SURFACES WITH AN INFINITE NUMBER OF ENDS

We prove the existence of an infinite family of periodic properly embedded minimal surfaces with an infinite number of annular ends. 25 refs., 29 figs.

The maximum principle at infinity for minimal surfaces in flat three manifolds.

Maximum principles are used as basic analytic tools for studying properties of functions defined on domains in R n and satisfying certain equations (e.g. elliptic). In general these maximum principles play a fundamental role in analysis on complete Riemannian manifolds, especially in the study of variational problems. For example, the well-known maximum principle for harmonic functions has had both a simplifying and unifying effect on the fields of harmonic and complex analysis. H. Hopf [18] gave an important general maximum principle for second order linear elliptic partial differential equations. The Hopf maximum principle easily yields a maximum principle for solutions of the minimal surface equation. In this context the principle states that if D c R 2 is a smooth connected domain and f~ ,f2 are two smooth functions on D that satisfy the minimal surface equation, then the difference fl -f2 cannot have an interior maximum or minimum unless the difference is constant. The maximum principle for minimal graphs gives rise to the following geometric result for minimal surfaces in Riemannian three-manifolds: IfM~ and ME are minimal surfaces in a Riemannian three-manifold that intersect at a common interior point p and M1 is on one side of M2 near p, then M~ intersects M2 in an open surface containing