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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 936826
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Abstract: Let $ G$ be a compact, connected subgroup of $ SO(n)$ acting on $ {{\mathbf{R}}^n}$, and let the action of $ G$ be polar. (Polar actions include the adjoint action of a Lie group $ H$ on the tangent space to the symmetric space $ G/H$ at the identity coset.) Let $ B$ be an $ (m - 1)$-dimensional submanifold without boundary, invariant under the action of $ G$, and lying in the union of the principal orbits of $ G$. It is shown that, if $ S$ is an area-minimizing integral current with boundary $ B$, then $ S$ is invariant under the action of $ G$. This result is extended to a larger class of boundaries, and to a class of parametric integrands including the area integrand.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0936826-4
PII: S 0002-9947(1988)0936826-4
Article copyright: © Copyright 1988 American Mathematical Society