^{*} Universität Bonn, Mathematisches Institut, Beringstrasse 1, 53115 Bonn, Germany; ^{} Department of Mathematics, University of Massachusetts, Amherst, MA 01003; and ^{} Department of Mathematics, University of Illinois, Urbana, IL 61801
Communicated by Richard M. Schoen, Stanford University, Stanford, CA, June 14, 1999 (received for review April 15, 1999)
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We announce the classification of complete almost embedded surfaces of constant mean curvature, with three ends and genus^{ }zero. They are classified by triples of points on the sphere whose^{ }distances are the asymptotic necksizes of the three^{ }ends.
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Surfaces that minimize area under a volume constraint have constant mean curvature (CMC); this condition can be expressed^{ }as a nonlinear partial differential equation. We are interested^{ }in complete CMC surfaces properly embedded in ^{3}; we rescale them to have mean curvature one. For technical reasons,^{ }we consider a slight generalization of embeddedness [introduced^{ }by Alexandrov (1)]: An immersed surface is almost embedded^{ }if it bounds a properly immersed threemanifold.
Alexandrov (1, 2) showed that the round sphere is the only compact almost embedded CMC surface. The next case to consider^{ }is that of finitetopology surfaces, homeomorphic to a compact^{ }surface with a finite number of points removed. A neighborhood^{ }of any of these punctures is called an end of the surface. The^{ }unduloids, CMC surfaces of revolution described by Delaunay (3),^{ }are genuszero examples with two ends. Each is a solution of an^{ }ordinary differential equation; the entire family is parametrized^{ }by the unduloid necksize, which ranges from zero (at the singular^{ }chain of spheres) to (at the^{ }cylinder).
Over the past decade there has been increasing understanding of finitetopology almost embedded CMC surfaces. Each end of^{ }such a surface is asymptotic to an unduloid (4). Meeks showed^{ }(5) there are no examples with a single end. The unduloids^{ }themselves are the only examples with two ends (4). Kapouleas^{ }(6) has constructed examples (near the limit of zero necksize)^{ }with any genus and any number of ends greater than^{ }two.
In this note we announce the classification of all almost embedded CMC surfaces with three ends and genus zero; we call these^{ }triunduloids (see Fig. 1). In light of the trousers decomposition^{ }for surfaces, triunduloids can be seen as the building blocks^{ }for more complicated almost embedded CMC surfaces (7). Our^{ }main result determines explicitly the moduli space of triunduloids^{ }with labeled ends, up to Euclidean motions. Because triunduloids^{ }are transcendental objects, and are not described by any ordinary^{ }differential equation, it is remarkable to have such a complete^{ }and explicit determination for their moduli space.

THEOREM. Triunduloids are classified by triples of distinct labeled points in the twosphere (up to rotations); the spherical^{ }distances of points in the triple are the necksizes of the unduloids^{ }asymptotic to the three ends. The moduli space of triunduloids^{ }is therefore homeomorphic to an open threeball.
The proof of the theorem has three parts. First we define the classifying map from triunduloids to spherical triples, and^{ }observe that it is proper; then we prove it is injective; and^{ }finally we show it is^{ }surjective.
To define the classifying map, we use the fact that any triunduloid has a reflection symmetry that decomposes the surface^{ }into mirrorimage halves (8). Each half is simply connected,^{ }so Lawson's construction (9) gives a conjugate cousin minimal^{ }surface in the threesphere. Using observations of Karcher (10),^{ }we find that its boundary projects under the Hopf map to the desired^{ }spherical triple. The composition of these steps defines our classifying^{ }map. It follows from curvature estimates (11) that the map is^{ }proper.
The injectivity of our classifying map is really a uniqueness result. We use the Hopf circle bundle to construct a trivial^{ }circle bundle over the disk representing the Lawson conjugate.^{ }Its total space is locally isometric to the threesphere, and^{ }so the circle action along the fibers is by isometries. The classifying^{ }triple determines the bundle up to isometries. Moreover, any conjugate^{ }surface with the same triple defines a minimal section of the^{ }bundle. Thus, we are in a situation familiar from minimal graphs,^{ }and we can apply a suitable maximum principle to deduce^{ }uniqueness.
Finally, we need an existence result showing that our classifying map is surjective. We depend on the fact (12) that the^{ }moduli space of CMC surfaces of genus g with k ends is locally^{ }a real analytic variety of (formal) dimension 3k 6. In particular,^{ }near a nondegenerate triunduloid, our moduli space has dimension^{ }three. We get such a nondegenerate triunduloid by using a nondegenerate^{ }minimal trinoid (13) in a recent construction by Mazzeo and^{ }Pacard.
To prove surjectivity of our classifying map, we use the fact (1416) that a threedimensional analytic variety can be triangulated,^{ }with each twosimplex meeting an even number of threesimplices.^{ }We then show that a proper, injective map from such a threecomplex^{ }to a connected threemanifold (here, the threeball) must be surjective^{ }as well. We use the standard lemma (17) that a proper, injective^{ }map from any space to a compactly generated space is a homeomorphism^{ }onto its image. Once surjectivity is known, this lemma is used^{ }once more to show that our classifying map is in fact a^{ }homeomorphism.
Note that our geometric picture of the triunduloid moduli space naturally explains necksize bounds for triunduloids. For instance,^{ }the symmetric triunduloids constructed previously (18) have^{ }three congruent ends, with necksize at most 2/3. This bound in^{ }the symmetric case can now be seen as the maximum side length^{ }for a spherical equilateral triangle. More generally, we have^{ }the^{ }following.
COROLLARY. The triple 0 < x, y, z can be the
necksizes of a triunduloid if and only if it satisfies the spherical
triangle^{ }inequalities:
Similar methods apply to genuszero surfaces with k > 3 ends, when those ends still have asymptotic axes in a common plane.^{ }The moduli space of such coplanar kunduloids can be understood^{ }as a covering of the space of spherical kgons. More general surfaces,^{ }without coplanar ends, will be more difficult to^{ }classify.
Acknowledgements 

This work was supported in part by the Deutsche Forschungsgemeinschaft through SFB 256 (K.G.B.), and by the National Science^{ }Foundation, under grants DMS9704949 (to R.B.K.) and DMS 9727859^{ }(to J.M.S.) and through GANG under grant DMS9626804.
Abbreviation 

CMC, constant mean curvature.
Footnotes 

^{§} To whom reprint requests should be addressed. Email: jms@math.uiuc.edu.
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