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.

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
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     title = {Doubling constant mean curvature tori in $S^3$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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     volume = {Ser. 5, 5},
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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://archive.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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl