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Annals of Global Analysis and Geometry

, Volume 36, Issue 3, pp 221–274 | Cite as

Gluing constructions amongst constant mean curvature hypersurfaces in \({\mathbb {S}^{n+1}}\)

  • Adrian ButscherEmail author
Original Paper

Abstract

Four constructions of constant mean curvature (CMC) hypersurfaces in \({\mathbb {S}^{n+1}}\) are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like ‘arms’. This last result extends Butscher and Pacard’s doubling construction for generalized Clifford tori of small mean curvature.

Keywords

Geometric analysis Constant mean curvature hypersurfaces Gluing Nonlinear elliptic partial differential equations 

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References

  1. 1.
    Abraham R., Marsden J.E., Ratiu T.: Manifolds, Tensor Analysis, and Applications, 2nd edn. Springer-Verlag, New York (1988)zbMATHGoogle Scholar
  2. 2.
    Alexandrov A.: A characteristic property of spheres. Ann. Mat. Pura Appl. 58, 303–315 (1962)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Brito F., Leite M.L.: A remark on rotational hypersurfaces of S n. Bull. Soc. Math. Belg. Sér. B 42(3), 303–318 (1990) MR 1081607 (91j:53028)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bryant, R.L.: Surfaces of mean curvature one in hyperbolic space, Astérisque 154–155 (1987), 12, 321–347, 353 (1988). Théorie des variétés minimales et applications (Palaiseau, 1983–1984)Google Scholar
  5. 5.
    Butscher A., Pacard F.: Doubling constant mean curvature tori in \({{\mathbb {S}}^3}\) . Ann. Sc. Norm. Super. Pisa. Cl. Sci. 5(4), 611–638 (2006)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Butscher A., Pacard F.: Generalized doubling constructions for constant mean curvature hypersurfaces in \({{\mathbb {S}}^{n+1}}\) . Ann. Glob. Anal. Geom. 32(2), 103–123 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Colares, G., Palmas, O.: Addendum to: “Complete rotation hypersurfaces with H k constant in space forms” [Bol. Soc. Brasil. Mat. (N.S.) 30(2), 139–161 (1999); mr1701417] by Palmas, Bull. Braz. Math. Soc. (N.S.) 39(1), 11–20 (2008)Google Scholar
  8. 8.
    Dorfmeister J., Pedit F., Wu H.: Weierstrass type representation of harmonic maps into symmetric spaces. Commun. Anal. Geom. 6(4), 633–668 (1998)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Haskins M., Kapouleas N.: Special Lagrangian cones with higher genus links. Invent. Math. 167(2), 223–294 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jleli M.: Moduli space theory of constant mean curvature hypersurfaces. Adv. Nonlinear Stud. 9, 29–68 (2009)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Jleli M., Pacard F.: An end-to-end construction for compact constant mean curvature surfaces. Pac. J. Math. 221(1), 81–108 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Kapouleas N.: Complete constant mean curvature surfaces in Euclidean three-space. Ann. Math. (2) 131(2), 239–330 (1990)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Kapouleas N.: Constant mean curvature surfaces constructed by fusing Wente tori. Invent. Math. 119(3), 443–518 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kapouleas N.: Complete embedded minimal surfaces of finite total curvature. J. Diff. Geom. 47(1), 95–169 (1997)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Kapouleas, N.: On desingularizing the intersections of minimal surfaces. In: Proceedings of the 4th International Congress of Geometry (Thessaloniki, 1996), Giachoudis-Giapoulis, Thessaloniki, 1997, pp. 34–41Google Scholar
  16. 16.
    Kenmotsu K.: Weierstrass formula for surfaces of prescribed mean curvature. Math. Ann. 245, 89–99 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Korevaar N., Kusner R., Solomon B.: The structure of complete embedded surfaces with constant mean curvature. J. Diff. Geom. 30, 465–503 (1989)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Kusner R., Mazzeo R., Pollack D.: The moduli space of complete embedded constant mean curvature surfaces. Geom. Funct. Anal. 6(1), 120–137 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Leite M.L.: Rotational hypersurfaces of space forms with constant scalar curvature. Manuscr. Math. 67(3), 285–304 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Mazzeo R., Pacard F.: Constant mean curvature surfaces with Delaunay ends. Commun. Anal. Geom. 9(1), 169–237 (2001)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Mazzeo R., Pacard F., Pollack D.: Connected sums of constant mean curvature surfaces in Euclidean 3 space. J. Reine Angew. Math. 536, 115–165 (2001)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Mazzeo, R.: Recent advances in the global theory of constant mean curvature surfaces. Noncompact problems at the intersection of geometry, analysis, and topology. Contemp. Math., Vol. 350, American Mathematical Society, Providence, RI, pp. 179–199 (2004)Google Scholar
  23. 23.
    Pacard F., Pimentel F.: Attaching handles to constant-mean-curvature-1 surfaces in hyperbolic 3-space. J. Inst. Math. Jussieu 3(3), 421–459 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Pacard, F., Rivière, T.: Linear and nonlinear aspects of vortices. The Ginzburg–Landau Model. Progress in Nonlinear Differential Equations and their Applications, vol. 39. Birkhäuser Boston Inc., Boston, MA (2000)Google Scholar
  25. 25.
    Palmas O.: Complete rotation hypersurfaces with H k constant in space forms. Bol. Soc. Brasil. Math. (N.S.) 30(2), 139–161 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sa Earp R., Toubiana E.: On the geometry of constant mean curvature one surfaces in hyperbolic space. Ill. J. Math. 45(2), 371–401 (2001)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Umehara M., Yamada K.: Complete surfaces of constant mean curvature 1 in the hyperbolic 3-space. Ann. Math. (2) 137(3), 611–638 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Wei, G.: Rotational hypersurfaces of the sphere. In: Proceedings of the Eleventh International Workshop on pp. 225–232Google Scholar
  29. 29.
    Wente H.: Counterexample to a conjecture of H. Hopf. Pac. Math. J. 191, 193–243 (1986)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Stanford UniversityStanfordUSA

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