Doubling constant mean curvature tori in S 3

Adrian Butscher; Frank Pacard

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 4, page 611-638
  • ISSN: 0391-173X

Abstract

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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.

How to cite

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Butscher, Adrian, and Pacard, Frank. "Doubling constant mean curvature tori in S$^{\textbf {3}}$." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.4 (2006): 611-638. <http://eudml.org/doc/243607>.

@article{Butscher2006,
abstract = {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.},
author = {Butscher, Adrian, Pacard, Frank},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {611-638},
publisher = {Scuola Normale Superiore, Pisa},
title = {Doubling constant mean curvature tori in S$^\{\textbf \{3\}\}$},
url = {http://eudml.org/doc/243607},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Butscher, Adrian
AU - Pacard, Frank
TI - Doubling constant mean curvature tori in S$^{\textbf {3}}$
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 4
SP - 611
EP - 638
AB - 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.
LA - eng
UR - http://eudml.org/doc/243607
ER -

References

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