Area-minimizing integral currents with boundaries invariant under polar actions

Author:
Julian C. Lander

Journal:
Trans. Amer. Math. Soc. **307** (1988), 419-429

MSC:
Primary 49F20; Secondary 53A10, 58E15

DOI:
https://doi.org/10.1090/S0002-9947-1988-0936826-4

MathSciNet review:
936826

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact, connected subgroup of acting on , and let the action of be polar. (Polar actions include the adjoint action of a Lie group on the tangent space to the symmetric space at the identity coset.) Let be an -dimensional submanifold without boundary, invariant under the action of , and lying in the union of the principal orbits of . It is shown that, if is an area-minimizing integral current with boundary , then is invariant under the action of . This result is extended to a larger class of boundaries, and to a class of parametric integrands including the area integrand.

**[1]**D. Bindschadler,*Invariant solutions to the oriented plateau problem of maximum codimension*, Trans. Amer. Math. Soc.**261**(1980), 439-461. MR**580897 (81k:49032)****[2]**J. E. Brothers,*Invariance of solutions to invariant parametric variational problems*, Trans. Amer. Math. Soc.**262**(1980), 159-179. MR**583850 (82h:49027)****[3]**J. Dadok,*Polar coordinates induced by actions of compact Lie groups*, Trans. Amer. Math. Soc.**288**(1985), 125-137. MR**773051 (86k:22019)****[4]**-, Personal communication.**[5]**H. Federer,*Geometric measure theory*, Springer-Verlag, New York, 1969. MR**0257325 (41:1976)****[6]**S. Helgason,*Differential geometry, Lie groups, and symmetric spaces*, Academic Press, New York, 1978. MR**514561 (80k:53081)****[7]**J. C. Lander,*Area-minimizing integral currents with boundaries invariant under polar actions*, Ph.D. Thesis, Massachusetts Institute of Technology, 1984.**[8]**H. B. Lawson, Jr.,*The equivariant Plateau problem and interior regularity*, Trans. Amer. Math. Soc.**173**(1972), 231-249. MR**0308905 (46:8017)****[9]**F. Morgan,*On finiteness of the number of stable minimal hypersurfaces with a fixed boundary*, Indiana Univ. Math. J.**35**(1985), 779-833. MR**865429 (88b:49059)****[10]**R. S. Palais and C.-L. Terng,*A general theory of canonical forms*, preprint. MR**876478 (88f:57069)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
49F20,
53A10,
58E15

Retrieve articles in all journals with MSC: 49F20, 53A10, 58E15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0936826-4

Article copyright:
© Copyright 1988
American Mathematical Society