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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Invariance of solutions to invariant parametric variational problems


Author: John E. Brothers
Journal: Trans. Amer. Math. Soc. 262 (1980), 159-179
MSC: Primary 49F22; Secondary 58A25
DOI: https://doi.org/10.1090/S0002-9947-1980-0583850-2
MathSciNet review: 583850
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Abstract: Let G be a compact Lie group of diffeomorphisms of a connected orientable manifold M of dimension $ n + 1$. Assume the orbits of highest dimension to be connected. Let $ \Psi $ be a convex positive even parametric integrand of degree n on M which is invariant under the action of G. Let T be a homologically $ \Psi $-minimizing rectifiable current of dimension n on M, and assume there exists a G-invariant rectifiable current $ T'$ which is homologous to T. It is shown that T is G-invariant provided $ \Psi $ satisfies a symmetry condition which makes it no less efficient for the tangent planes of T to lie along the orbits. This condition is satisfied by the area integrand in case G is a group of isometries of a Riemannian metric on M. Consequently, one obtains the corollary that if a rectifiable current T is a solution to the n-dimensional Plateau problem in M with G-invariant boundary $ \partial T$, and if $ \partial T$ bounds a G-invariant rectifiable current $ T'$ such that $ T - T'$ is a boundary, then T is G-invariant. An application to the Plateau problem in $ {{\textbf{S}}^3}$ is given.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0583850-2
Keywords: Parametric integrand, G-invariant current, homologically $ \Psi $-minimizing rectifiable current, Plateau problem
Article copyright: © Copyright 1980 American Mathematical Society