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 . Assume the orbits of highest dimension to be connected. Let 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 -minimizing rectifiable current of dimension *n* on *M*, and assume there exists a *G*-invariant rectifiable current which is homologous to *T*. It is shown that *T* is *G*-invariant provided 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 , and if bounds a *G*-invariant rectifiable current such that is a boundary, then *T* is *G*-invariant. An application to the Plateau problem in is given.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0583850-2

Keywords:
Parametric integrand,
*G*-invariant current,
homologically -minimizing rectifiable current,
Plateau problem

Article copyright:
© Copyright 1980
American Mathematical Society