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Constant mean curvature hypersurfaces in $${{\mathbb S}^{n+1}}$$ by gluing spherical building blocks

Constant mean curvature hypersurfaces in $${{\mathbb S}^{n+1}}$$ by gluing spherical building blocks The techniques developed by Butscher (Gluing constructions amongst constant mean curvature hypersurfaces of $${{\mathbb S}^{n+1}}$$ ) for constructing constant mean curvature (CMC) hypersurfaces in $${{\mathbb S}^{n+1}}$$ by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by gluing the initial configuration together can be perturbed into an exactly CMC hypersurface only when certain global geometric conditions are met. These balancing conditions are analogous to those that must be satisfied in the “classical” context of gluing constructions of CMC hypersurfaces in Euclidean space, although they are more restrictive in the $${{\mathbb S}^{n+1}}$$ case. An example of an initial configuration is given which demonstrates this fact; and another example of an initial configuration is given which possesses no symmetries at all. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Constant mean curvature hypersurfaces in $${{\mathbb S}^{n+1}}$$ by gluing spherical building blocks

Butscher, Adrian
Mathematische Zeitschrift , Volume 263 (1) – Aug 26, 2008
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References (12)

Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer-Verlag
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-008-0407-5
Publisher site
See Article on Publisher Site

Abstract

The techniques developed by Butscher (Gluing constructions amongst constant mean curvature hypersurfaces of $${{\mathbb S}^{n+1}}$$ ) for constructing constant mean curvature (CMC) hypersurfaces in $${{\mathbb S}^{n+1}}$$ by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by gluing the initial configuration together can be perturbed into an exactly CMC hypersurface only when certain global geometric conditions are met. These balancing conditions are analogous to those that must be satisfied in the “classical” context of gluing constructions of CMC hypersurfaces in Euclidean space, although they are more restrictive in the $${{\mathbb S}^{n+1}}$$ case. An example of an initial configuration is given which demonstrates this fact; and another example of an initial configuration is given which possesses no symmetries at all.

Journal

Mathematische Zeitschrift – Springer Journals

Published: Aug 26, 2008

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