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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Invariant solutions to the oriented Plateau problem of maximal codimension


Author: David Bindschadler
Journal: Trans. Amer. Math. Soc. 261 (1980), 439-462
MSC: Primary 49F22; Secondary 58E12
MathSciNet review: 580897
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Abstract: The principal result gives conditions which imply that a solution to the Plateau problem inherits the symmetries of its boundary. Specifically, let G be a compact connected Lie subgroup of $ {\text{SO}}(n)$. Assume the principal orbits have dimension m, there are no exceptional orbits and the distribution of $ (n\, - \,m)$-planes orthogonal to the principal orbits is involutive. We show that if B is a finite sum of oriented principal orbits, then every absolutely area minimizing current with boundary B is invariant.

As a consequence of the methods used, the above Plateau problems are shown to be equivalent to 1-dimensional variational problems in the orbit space. Some results concerning invariant area minimizing currents in Riemannian manifolds are also obtained.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0580897-7
PII: S 0002-9947(1980)0580897-7
Keywords: Invariant solutions, area minimizing, mass minimizing, oriented Plateau problem, variational problem, lifting currents, projecting currents, involutive distribution, group action, orbit space
Article copyright: © Copyright 1980 American Mathematical Society