The Clifford tori in 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.
@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},
pages = {611--638},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 5},
number = {4},
year = {2006},
zbl = {1170.53303},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2006_5_5_4_611_0/}
}
TY - JOUR AU - Butscher, Adrian AU - Pacard, Frank TI - Doubling constant mean curvature tori in $S^3$ JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2006 SP - 611 EP - 638 VL - 5 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2006_5_5_4_611_0/ LA - en ID - ASNSP_2006_5_5_4_611_0 ER -
%0 Journal Article %A Butscher, Adrian %A Pacard, Frank %T Doubling constant mean curvature tori in $S^3$ %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2006 %P 611-638 %V 5 %N 4 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2006_5_5_4_611_0/ %G en %F 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] , and , “Manifolds, Tensor Analysis, and Applications”, second ed., Springer-Verlag, New York, 1988. | MR | Zbl
[2] , A characteristic property of spheres, Ann. Mat. Pura Appl. 58 (1962), 303-315. | MR | Zbl
[3] , and , Weierstrass type representation of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), 633-668. | MR | Zbl
[4] , New surfaces of constant mean curvature, Math. Z. 214 (1993), 527-565. | MR | Zbl
[5] and , An end-to-end construction for compact constant mean curvature surfaces, Pacific J. Math. 221 (2005), 81-108. | MR | Zbl
[6] , Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. 131 (1990), 239-330. | MR | Zbl
[7] , Constant mean curvature surfaces constructed by fusing Wente tori, Invent. Math. 119 (1995), 443-518. | MR | Zbl
[8] , 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] , The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscripta Math. 64 (1989), 291-357. | MR | Zbl
[10] , Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979), 89-99. | MR | Zbl
[11] , and , The moduli space of complete embedded constant mean curvature surfaces, Geom. Funct. Anal. 6 (1996), 120-137. | MR | Zbl
[12] , Complete minimal surfaces in , Ann. of Math. 92 (1970), 335-374. | MR | Zbl
[13] and , Constant mean curvature surfaces with Delaunay ends, Comm. Anal. Geom. 9 (2001), 169-237. | MR | Zbl
[14] and , Bifurcating nodoids, In: “Topology and Geometry: Commemorating SISTAG”, Contemp. Math., Vol. 314, Amer. Math. Soc., Providence, RI, 2002, 169-186. | MR | Zbl
[15] , and , Connected sums of constant mean curvature surfaces in Euclidean space, J. Reine Angew. Math. 536 (2001), 115-165. | MR | Zbl
[16] , 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] and , On the classification of constant mean curvature tori, Ann. of Math. 130 (1989), 407-451. | MR | Zbl
[18] and , Equivariant minimax and minimal surfaces in geometric three-manifolds, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 303-309. | MR | Zbl
[19] , Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere, Math. Z. 226 (1997), 127-146. | MR | Zbl
[20] , Counterexample to a conjecture of H. Hopf, Pacific J. Math. 191 (1986), 193-243. | MR | Zbl
[21] , Minimal surfaces in and constructed by gluing, Proc. of the International Workshop on Differential Geometry (Taegu), Kyungpook National University, 2003, 183-192. | MR | Zbl
