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

DOI:
https://doi.org/10.1090/S0002-9947-1980-0580897-7

MathSciNet review:
580897

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Abstract | References | Similar Articles | Additional Information

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 . Assume the principal orbits have dimension *m*, there are no exceptional orbits and the distribution of -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.

**[B]**D. E. Bindschadler,**Invariant and singular solutions to the Plateau problem in Riemannian manifolds**, Ph.D. Thesis, Indiana University, 1976.**[BG]**G. E. 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]**-,*A characterization of integral currents*, Trans. Amer. Math. Soc.**150**(1970), 301-325. MR**0266125 (42:1034)****[BJ3]**-,*The structure of solutions to Plateau's problem in the n-sphere*, J. Differential Geometry**11**(1976), 387-400. MR**0435995 (55:8946)****[Fl]**H. Federer,*Geometric measure theory*, Springer-Verlag, Berlin and New York, 1969. MR**0257325 (41:1976)****[F2]**-,*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)****[L]**H. B. Lawson,*The equivariant Plateau problem and interior regularity*, Trans. Amer. Math. Soc.**173**(1972), 231-249. MR**0308905 (46:8017)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0580897-7

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