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 | 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]**Glen E. Bredon,*Introduction to compact transformation groups*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR**0413144****[BJ1]**John E. Brothers,*Integral geometry in homogeneous spaces*, Trans. Amer. Math. Soc.**124**(1966), 480–517. MR**0202099**, 10.1090/S0002-9947-1966-0202099-9**[BJ2]**John E. Brothers,*A characterization of integral currents*, Trans. Amer. Math. Soc.**150**(1970), 301–325. MR**0266125**, 10.1090/S0002-9947-1970-0266125-4**[BJ3]**John E. Brothers,*The structure of solutions to Plateau’s problem in the 𝑛-sphere*, J. Differential Geometry**11**(1976), no. 3, 387–400. MR**0435995****[Fl]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[F2]**Herbert Federer,*Real flat chains, cochains and variational problems*, Indiana Univ. Math. J.**24**(1974/75), 351–407. MR**0348598****[FF]**Herbert Federer and Wendell H. Fleming,*Normal and integral currents*, Ann. of Math. (2)**72**(1960), 458–520. MR**0123260****[L]**H. Blaine Lawson Jr.,*The equivariant Plateau problem and interior regularity*, Trans. Amer. Math. Soc.**173**(1972), 231–249. MR**0308905**, 10.1090/S0002-9947-1972-0308905-4

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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