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End-to-end gluing of constant mean curvature hypersurfaces
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 4, p. 717-737

It was observed by R. Kusner and proved by J. Ratzkin that one can connect together two constant mean curvature surfaces having two ends with the same Delaunay parameter. This gluing procedure is known as a “end-to-end connected sum”. In this paper we generalize, in any dimension, this gluing procedure to construct new constant mean curvature hypersurfaces starting from some known hypersurfaces.

Il a été observé par R. Kusner et prouvé par J. Ratzkin qu’on peut recoller ensemble deux surfaces à courbure moyenne constante ayant deux bouts de même paramètre de Delaunay. Cette procédure de recollement est connu comme « somme connexe bout-à-bout ». Dans ce papier, nous donnons une généralisation de cette construction en dimension quelconque dans le but de construire des nouvelles hypersurfaces à courbure moyenne constante à partir des hypersurfaces connues.

Received : 2008-02-29
Accepted : 2008-11-21
Published online : 2010-01-04
DOI : https://doi.org/10.5802/afst.1222
@article{AFST_2009_6_18_4_717_0,
     author = {Mohamed Jleli},
     title = {End-to-end gluing of constant mean curvature hypersurfaces},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 18},
     number = {4},
     year = {2009},
     pages = {717-737},
     doi = {10.5802/afst.1222},
     mrnumber = {2590386},
     zbl = {1206.53010},
     language = {en},
     url={afst.centre-mersenne.org/item/AFST_2009_6_18_4_717_0/}
}
Jleli, Mohamed. End-to-end gluing of constant mean curvature hypersurfaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 4, pp. 717-737. doi : 10.5802/afst.1222. https://afst.centre-mersenne.org/item/AFST_2009_6_18_4_717_0/

[1] Delaunay (C.).— Sur la surface de révolution dont la courbure moyenne est constante, Jour. de Mathématique, 6, p. 309-320 (1841).

[2] Eells (J.).— The surfaces of Delaunay , Math. Intelligencer 9, no.1, p. 53-57 (1987). | MR 869541 | Zbl 0605.53002

[3] Fakhi (S.) and Pacard (F.).— Existence result for minimal hypersurfaces with prescribed finite number of planar end , Manuscripta Mathematica, vol 103, issu 4, p. 465-512 (2000). | MR 1811769 | Zbl 0992.53011

[4] Hsiang (W. Y.) and Yu (W. C.).— A generalization of a theorem of Delaunay, J. Differ. Geom. 16, No. 2, p. 161-177 (1981). | MR 638783 | Zbl 0504.53044

[5] Jleli (M.).— Moduli space theory of constant mean curvature hypersurfaces. Journal of Advanced Nonlinear Studies, 9 p. 29-68 (2009). | MR 2473148 | Zbl pre05567718

[6] Jleli (M.) and Pacard (F.).— Construction of constant mean curvature hypersurfaces with prescribed finite number of Delaunay end. To appear.

[7] Jleli (M.) and Pacard (F.).— An end-to-end construction for compact constant mean curvature surfaces Pacific Journal of Mathematics Vol. 221, No. 1, p. 81-108 (2005). | MR 2194146 | Zbl 1110.53043

[8] Kapouleas (N.).— Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131, p. 239-330 (1990). | MR 1043269 | Zbl 0699.53007

[9] Kapouleas (N.).— Compact constant mean curvature surfaces in Euclidean three-space, J. Differ. Geom. 33, No. 3, p. 683-715 (1991). | MR 1100207 | Zbl 0727.53063

[10] Kapouleas (N.).— Constant mean curvature surfaces constructed by fusing Went tori, Invent. Math. 119, p. 443-518 (1995). | MR 1317648 | Zbl 0840.53005

[11] Katsuei (K.).— Surfaces of revolution with prescribed mean curvature. Tohoku. Math. J ser 32, p. 147-153 (1980). | MR 567837 | Zbl 0431.53005

[12] Katsuei (K.).— Surfaces of revolution with prescribed mean curvature. Tohoku. Math. J ser 32, p. 147-153 (1980). | MR 567837 | Zbl 0431.53005

[13] Kusner (R.).— Bubbles conservations laws and balanced diagram , Geometric analysis and Computer graphics, (1991) 120-137. Springer-Verlag. | MR 1081331

[14] Kusner (R.), Mazzeo (R.) and Pollack (D.).— The moduli spaces of complete embeeded constant mean curvature surfaces , Geom. Funct. Anal. 6, p. 120-137 (1996). | MR 1371233 | Zbl 0966.58005

[15] Mazzeo (R.) and Pacard (F.).— Constant mean curvature surfaces with Delaunay ends, Comm. Anal. Geom. 9 No. 1 p. 169-237 (2001). | MR 1807955 | Zbl 1005.53006

[16] Mazzeo (R.), Pacard (F.) and Pollack (D.).— Connected sums of constant mean curvature surfaces in Euclidiean 3 space, J.Reine Ang.Math. 536, p. 115.165 (2001). | MR 1837428 | Zbl 0972.53010

[17] Mazzeo (R.), Pacard (F.) and Pollack (D.).— The conformal theory of Alexandrov embedded constant mean curvature surfaces in 3 , in Global theory of minimal surfaces, edited by D. Hoffman, Clay Mathematics Proceedings 2, Amer. Math. Soc, Providence, p. 525-559 (2005). | MR 2167275 | Zbl 1101.53006

[18] Mazzeo (R.), Pollack (D.) and Uhlenbeck (K.).— Moduli spaces of singular Yammabe metrics , J. Amer. Math. 9, p. 303-344 (1996). | MR 1356375 | Zbl 0849.58012

[19] Ratzkin (J.).— An end-to-end gluing construction for surfaces of constant mean curvature, PHD Thesis, University of Washington (2001).

[20] Rosenberg (H.).— Hypersurfaces of constant mean curvature in space forms, Bull. Sc. math, série 2, 117, p. 211-239 (1993). | Zbl 0787.53046