The moduli space of complete embedded constant mean curvature surfaces

Kusner, Robert B.; Mazzeo, Rafe; Pollack, Daniel

doi:10.1007/bf02246769

Cited by 65 publications

(108 citation statements)

References 13 publications

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“…The general analysis of the moduli space of Alexandrov embedded CMC surfaces was considered by the first author, Kusner and Pollack [9] (essentially merely translating the analogous results in [13] for the singular Yamabe problem). The results there easily translate to the slightly more general moduli space Mg^k (the elements of which are not required to be Alexandrov embedded, but simply have Delaunay ends) without any difficulty.…”

Section: G K = ML K L>ml K Um^kmentioning

confidence: 99%

Constant mean curvature surfaces with Delaunay ends

Mazzeo¹,

Pacard²

2001

Communications in Analysis and Geometry

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132

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Section: G K = ML K L>ml K Um^kmentioning

confidence: 99%

Constant mean curvature surfaces with Delaunay ends

Mazzeo¹,

Pacard²

2001

Communications in Analysis and Geometry

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132

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“…This construction is based on two important tools which has been developed for the understanding of complete noncompact constant mean curvature surfaces. The first is the moduli space theory which is developed in [15] and the second is the end-to-end gluing of constant mean curvature surfaces developed in [18].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Recently, in [5], we generalized the result of [15]. In order to find a description of the structure of the set M k near any nondegenerate element, we prove a maximum principle for L Dτ .…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

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Construction of compact constant mean curvature hypersurfaces with topology

Jleli¹

2012

Ann. inst. Fourier

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“…We depend on the fact [13] 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.…”

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confidence: 99%

Constant Mean Curvature Surfaces with Cylindrical Ends

Große-Brauckmann

Kusner

Sullivan

1998

Mathematical Visualization

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Abstract. 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.Surfaces which 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 R 3 ; we rescale them to have mean curvature one. For technical reasons, we consider a slight generalization of embeddedness (introduced by Alexandrov [2]): An immersed surface is almost embedded if it bounds a properly immersed three-manifold.Alexandrov [1,2] showed that the round sphere is the only compact almost embedded cmc surface. The next case to consider is that of finite-topology 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 [4], are genus-zero 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 π (at the cylinder).Over the past decade there has been increasing understanding of finite-topology almost embedded cmc surfaces. Each end of such a surface is asymptotic to an unduloid [12]. Meeks [16] showed there are no examples with a single end. The unduloids themselves are the only examples with two ends [12]. Kapouleas [10] has constructed examples (near the zero-necksize limit) 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 figure). In light of the trousers decomposition for surfaces, triunduloids can be seen as the building blocks for more complicated almost embedded cmc surfaces [8]. Our main result determines explicitly the moduli space of triunduloids with labelled ends, up to Euclidean motions. Since 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.

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The moduli space of complete embedded constant mean curvature surfaces

Cited by 65 publications

References 13 publications

Constant mean curvature surfaces with Delaunay ends

Constant mean curvature surfaces with Delaunay ends

Construction of compact constant mean curvature hypersurfaces with topology

Constant Mean Curvature Surfaces with Cylindrical Ends

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