Construction of compact constant mean curvature hypersurfaces with topology
[Construction des hypersurfaces compactes à courbure moyenne constante qui ont une topologie non triviale]
Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 245-276.

Dans cet article, nous expliquons comment la méthode de construction dite “recollement des surfaces bout-à-bout” avec des resultats sur l’ensemble des hypersurfaces complètes non compactes à courbure moyenne constante qui ont un nombre fini de bouts de type Delaunay peuvent être utilisées pour construire des nouvelles familles d’hypersurfaces compactes à courbure moyenne constante qui ont une topologie non triviale.

In this paper, we explain how the end-to-end construction together with the moduli space theory can be used to produce compact constant mean curvature hypersurfaces with nontrivial topology. For the sake of simplicity, the hypersurfaces we construct have a large group of symmetry but the method can certainly be used to provide many more examples with less symmetries.

DOI : 10.5802/aif.2705
Classification : 35J05, 53A07, 53C21
Keywords: Mean curvature, Compact hypersurface
Mot clés : Courbure moyenne, hypersurface compacte
Jleli, Mohamed 1

1 Department of Mathematics College of Science King Saud University PO. Box 2455 Riyadh 11451 (Saudi Arabia)
@article{AIF_2012__62_1_245_0,
     author = {Jleli, Mohamed},
     title = {Construction of compact constant mean curvature hypersurfaces with topology},
     journal = {Annales de l'Institut Fourier},
     pages = {245--276},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {1},
     year = {2012},
     doi = {10.5802/aif.2705},
     zbl = {1250.53008},
     mrnumber = {2986271},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2705/}
}
TY  - JOUR
AU  - Jleli, Mohamed
TI  - Construction of compact constant mean curvature hypersurfaces with topology
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 245
EP  - 276
VL  - 62
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2705/
DO  - 10.5802/aif.2705
LA  - en
ID  - AIF_2012__62_1_245_0
ER  - 
%0 Journal Article
%A Jleli, Mohamed
%T Construction of compact constant mean curvature hypersurfaces with topology
%J Annales de l'Institut Fourier
%D 2012
%P 245-276
%V 62
%N 1
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2705/
%R 10.5802/aif.2705
%G en
%F AIF_2012__62_1_245_0
Jleli, Mohamed. Construction of compact constant mean curvature hypersurfaces with topology. Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 245-276. doi : 10.5802/aif.2705. http://www.numdam.org/articles/10.5802/aif.2705/

[1] Delaunay, C. Sur la surface de révolution dont la courbure moyenne est constante, Jour. de Mathématique (1841) no. 6, pp. 309-320 | EuDML | Numdam

[2] Große-Brauckmann, Karsten New surfaces of constant mean curvature, Math. Z., Volume 214 (1993) no. 4, pp. 527-565 | DOI | EuDML | MR | Zbl

[3] Große-Brauckmann, Karsten; Kusner, Robert B.; Sullivan, John M. Triunduloids: embedded constant mean curvature surfaces with three ends and genus zero, J. Reine Angew. Math., Volume 564 (2003), pp. 35-61 | DOI | MR | Zbl

[4] Jleli, Mohamed End-to-end gluing of constant mean curvature hypersurfaces, Ann. Fac. Sci. Toulouse Math. (6), Volume 18 (2009) no. 4, pp. 717-737 | DOI | EuDML | Numdam | MR | Zbl

[5] Jleli, Mohamed Moduli space theory of constant mean curvature hypersurfaces, Adv. Nonlinear Stud., Volume 9 (2009) no. 1, pp. 29-68 | MR | Zbl

[6] Jleli, Mohamed Symmetry-breaking for immersed constant mean curvature hypersurfaces, Adv. Nonlinear Stud., Volume 9 (2009) no. 2, pp. 243-261 | MR | Zbl

[7] Jleli, Mohamed; Pacard, Frank Construction of constant mean curvature hypersurfaces with prescribed finite number of Delaunay end (To appear) | Zbl

[8] Jleli, Mohamed; Pacard, Frank An end-to-end construction for compact constant mean curvature surfaces, Pacific J. Math., Volume 221 (2005) no. 1, pp. 81-108 | DOI | MR | Zbl

[9] Kapouleas, Nicolaos Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2), Volume 131 (1990) no. 2, pp. 239-330 | DOI | MR | Zbl

[10] Kapouleas, Nikolaos Constant mean curvature surfaces constructed by fusing Wente tori, Invent. Math., Volume 119 (1995) no. 3, pp. 443-518 | DOI | MR | Zbl

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

[12] Kilian, Martin; McIntosh, Ian; Schmitt, Nicholas New constant mean curvature surfaces, Experiment. Math., Volume 9 (2000) no. 4, pp. 595-611 http://projecteuclid.org/getRecord?id=euclid.em/1045759525 | DOI | MR

[13] Korevaar, Nicholas J.; Kusner, Rob; Solomon, Bruce The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom., Volume 30 (1989) no. 2, pp. 465-503 http://projecteuclid.org/getRecord?id=euclid.jdg/1214443598 | MR | Zbl

[14] Kusner, Rob Bubbles, conservation laws, and balanced diagrams, Geometric analysis and computer graphics (Berkeley, CA, 1988) (Math. Sci. Res. Inst. Publ.), Volume 17, Springer, New York, 1991, pp. 103-108 | MR

[15] Kusner, Rob; Mazzeo, R.; Pollack, D. The moduli space of complete embedded constant mean curvature surfaces, Geom. Funct. Anal., Volume 6 (1996) no. 1, pp. 120-137 | DOI | MR | Zbl

[16] Mazzeo, Rafe; Pacard, Frank Constant mean curvature surfaces with Delaunay ends, Comm. Anal. Geom., Volume 9 (2001) no. 1, pp. 169-237 | MR

[17] Mazzeo, Rafe; Pollack, Daniel; Uhlenbeck, Karen Moduli spaces of singular Yamabe metrics, J. Amer. Math. Soc., Volume 9 (1996) no. 2, pp. 303-344 | DOI | MR | Zbl

[18] Ratzkin, J. An ened-to-end gluing construction for surfaces of constant mean curvature, University of Washington (2001) (Ph. D. Thesis)

[19] Rosenberg, Harold Hypersurfaces of constant curvature in space forms, Bull. Sci. Math., Volume 117 (1993) no. 2, pp. 211-239 | MR | Zbl

[20] Wente, Henry C. Counterexample to a conjecture of H. Hopf, Pacific J. Math., Volume 121 (1986) no. 1, pp. 193-243 http://projecteuclid.org/getRecord?id=euclid.pjm/1102702809 | MR | Zbl

Cité par Sources :