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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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/item/ASNSP_2006_5_5_4_611_0/