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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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  • [BD] D. Bindschadler, Invariant solutions to the oriented Plateau problem in a class of Riemannian manifolds, Trans. Amer. Math. Soc. (to appear). MR 580897 (81k:49032)
  • [BG] G. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR 0413144 (54:1265)
  • [BJ1] J. E. Brothers, Integral geometry in homogeneous spaces, Trans. Amer. Math. Soc. 124 (1966), 480-517. MR 0202099 (34:1973)
  • [BJ2] -, The structure of solutions to Plateau's problem in the n-sphere, J. Differential Geometry 11 (1976), 387-400. MR 0435995 (55:8946)
  • [F1] H. Federer, Geometric measure theory, Springer-Verlag, Berlin and New York, 1969. MR 0257325 (41:1976)
  • [F2] -, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767-771. MR 0260981 (41:5601)
  • [F3] -, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974), 351-407. MR 0348598 (50:1095)
  • [FF] H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458-520. MR 0123260 (23:A588)
  • [HL] W. Y. Hsiang and H. B. Lawson, Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geometry 5 (1971), 1-38. MR 0298593 (45:7645)
  • [KN] S. Kobayashi and K. Nomizu, Foundations of differential geometry. I, Interscience, New York, 1963. MR 0152974 (27:2945)
  • [L1] H. B. Lawson, Jr., Complete minimal surfaces in $ {{\textbf{S}}^3}$, Ann. of Math. (2) 92 (1970), 335-374. MR 0270280 (42:5170)
  • [L2] -, The equivariant Plateau problem and interior regularity, Trans. Amer. Math. Soc. 173 (1972), 231-249. MR 0308905 (46:8017)
  • [SSA] R. Schoen, L. Simon and F. J. Almgren, Jr., Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals, Acta Math 139 (1977), 217-265. MR 0467476 (57:7333)

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Additional Information

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

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