Doubling constant mean curvature tori in S 3
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, p. 611-638
The Clifford tori in S 3 constitute a one-parameter family of flat, two-dimensional, constant mean curvature (CMC) submanifolds. This paper demonstrates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small catenoidal bridges can be constructed by perturbative PDE methods. That is, one can create a submanifold that has almost everywhere constant mean curvature by gluing a re-scaled catenoid into the neighbourhood of each point of a sub-lattice of the Clifford torus; and then one can show that a constant mean curvature perturbation of this submanifold does exist.
Classification:  53A10,  58J10
@article{ASNSP_2006_5_5_4_611_0,
     author = {Butscher, Adrian and Pacard, Frank},
     title = {Doubling constant mean curvature tori in $S^3$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {4},
     year = {2006},
     pages = {611-638},
     zbl = {1170.53303},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2006_5_5_4_611_0}
}
Butscher, Adrian; Pacard, Frank. Doubling constant mean curvature tori in $S^3$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, pp. 611-638. http://www.numdam.org/item/ASNSP_2006_5_5_4_611_0/

[1] R. Abraham, J. E. Marsden and T. Ratiu, “Manifolds, Tensor Analysis, and Applications”, second ed., Springer-Verlag, New York, 1988. | MR 960687 | Zbl 0875.58002

[2] A. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. 58 (1962), 303-315. | MR 143162 | Zbl 0107.15603

[3] J. Dorfmeister, F. Pedit and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), 633-668. | MR 1664887 | Zbl 0932.58018

[4] K. Große-Brauckmann, New surfaces of constant mean curvature, Math. Z. 214 (1993), 527-565. | MR 1248112 | Zbl 0806.53005

[5] M. Jleli and F. Pacard, An end-to-end construction for compact constant mean curvature surfaces, Pacific J. Math. 221 (2005), 81-108. | MR 2194146 | Zbl 1110.53043

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

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

[8] N. Kapouleas, Constant mean curvature surfaces in Euclidean spaces, Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Zürich, 1994, Basel, Birkhäuser, 1995, 481-490. | MR 1403948 | Zbl 0841.53006

[9] H. Karcher, The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscripta Math. 64 (1989), 291-357. | MR 1003093 | Zbl 0687.53010

[10] K. Kenmotsu, Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979), 89-99. | MR 552581 | Zbl 0402.53002

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

[12] H. Blaine Lawson, Complete minimal surfaces in S 3 , Ann. of Math. 92 (1970), 335-374. | MR 270280 | Zbl 0205.52001

[13] R. Mazzeo and F. Pacard, Constant mean curvature surfaces with Delaunay ends, Comm. Anal. Geom. 9 (2001), 169-237. | MR 1807955 | Zbl 1005.53006

[14] R. Mazzeo and F. Pacard, Bifurcating nodoids, In: “Topology and Geometry: Commemorating SISTAG”, Contemp. Math., Vol. 314, Amer. Math. Soc., Providence, RI, 2002, 169-186. | MR 1941630 | Zbl 1032.53002

[15] R. Mazzeo, F. Pacard and D. Pollack, Connected sums of constant mean curvature surfaces in Euclidean 3 space, J. Reine Angew. Math. 536 (2001), 115-165. | MR 1837428 | Zbl 0972.53010

[16] R. Mazzeo, Recent advances in the global theory of constant mean curvature surfaces, In: “Noncompact Problems at the Intersection of Geometry, Analysis, and Topology”, Contemp. Math., Vol. 350, Amer. Math. Soc., Providence, RI, 2004, 179-199. | MR 2082398 | Zbl 1075.53008

[17] U. Pinkall and I. Sterling, On the classification of constant mean curvature tori, Ann. of Math. 130 (1989), 407-451. | MR 1014929 | Zbl 0683.53053

[18] J. T. Pitts and J. H. Rubinstein, Equivariant minimax and minimal surfaces in geometric three-manifolds, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 303-309. | MR 940493 | Zbl 0665.49034

[19] M. Ritoré, Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere, Math. Z. 226 (1997), 127-146. | MR 1472144 | Zbl 0882.53010

[20] H. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 191 (1986), 193-243. | MR 815044 | Zbl 0586.53003

[21] S.-D. Yang, Minimal surfaces in 𝐄 3 and 𝐒 3 (1) constructed by gluing, Proc. of the 7 th International Workshop on Differential Geometry (Taegu), Kyungpook National University, 2003, 183-192. | MR 1966441 | Zbl 1035.53086