The moduli space of complete embedded constant mean curvature surfaces. (English) Zbl 0966.58005

Summary: Given \(Q > 1\), we construct an Ahlfors \(Q\)-regular space that admits a weak \((1,1)\)-Poincaré inequality.


58D10 Spaces of embeddings and immersions
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text: DOI arXiv EuDML


[1] J. Berglund, Ph.D. Thesis, University of Massachusetts at Amherst, 1996.
[2] C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constant, J. Math. Pure Appl. 6 (1841), 309–320.
[3] K. Grosse-Brauckmann, New surfaces of constant mean curvature, Math. Z. 214 (1993), 527–565. · Zbl 0806.53005 · doi:10.1007/BF02572424
[4] K. Grosse-Brauckmann, R. Kusner, On the moduli spaces of embedded constant mean curvature surfaces with three or four ends, GANG Preprint. · Zbl 0980.53011
[5] W.Y. Hsiang, On generalization of theorems of A.D. Alexandrov and C. Delaunay on hypersurfaces of constant mean curvature, Duke Math. Jour. 49 (1982), 485–496. · Zbl 0496.53006 · doi:10.1215/S0012-7094-82-04927-4
[6] N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. 131 (1990), 239–330. · Zbl 0699.53007 · doi:10.2307/1971494
[7] N. Korevaar, R. Kusner, The global structure of constant mean curvature surfaces, Invent. Math. 114 (1993), 311–332. · Zbl 0803.53040 · doi:10.1007/BF01232673
[8] N. Korevaar, R. Kusner, W. Meeks III, B. Solomon, Constant mean curvature surfaces in hyperbolic space, Amer. J. Math. 114 (1992), 1–43. · Zbl 0757.53032 · doi:10.2307/2374738
[9] N. Korevaar, R. Kusner, B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Diff. Geom. 30 (1989), 465–503. · Zbl 0726.53007
[10] R. Mazzeo, D. Pollack, K. Uhlenbeck, Moduli spaces of singular Yamabe metrics, To appear, Jour. Amer. Math. Soc. · Zbl 0849.58012
[11] R. Mazzeo, D. Pollack, K. Uhlenbeck, Connected sum constructions for constant scalar curvature metrics, Preprint. · Zbl 0866.58069
[12] W. Meeks III, The topology and geometry of embedded surfaces of constant mean curvature, J. Differential Geometry 27 (1988), 539–552. · Zbl 0617.53007
[13] R. Melrose, The Atiyah-Patodi-Singer Index Theorem, AK Peters Ltd., Wellesley, MA, 1993. · Zbl 0796.58050
[14] R. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure and Appl. Math. XLI (1988), 317–392. · Zbl 0674.35027 · doi:10.1002/cpa.3160410305
[15] L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Annals of Math. 118 (1983), 525–571. · Zbl 0549.35071 · doi:10.2307/2006981
[16] C. Taubes, Gauge theory on asymptotically periodic 4-manifolds, J. Differential Geometry 25 (1987), 363–430. · Zbl 0615.57009
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