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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The equivariant Plateau problem and interior regularity

Author: H. Blaine Lawson
Journal: Trans. Amer. Math. Soc. 173 (1972), 231-249
MSC: Primary 49F10; Secondary 53A10
MathSciNet review: 0308905
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Abstract: Let $ M \subset {{\text{R}}^n}$ be a compact submanifold of Euclidean space which is invariant by a compact group $ G \subset SO(n)$. When $ \dim (M) = n - 2$, it is shown that there always exists a solution to the Plateau problem for $ M$ which is invariant by $ G$ and, furthermore, that uniqueness of this solution among $ G$-invariant currents implies uniqueness in general. This result motivates the subsequent study of the Plateau problem for $ M$ within the class of $ G$-invariant integral currents. It is shown that this equivariant problem reduces to the study of a corresponding Plateau problem in the orbit space $ {\text{R}}/G$ where, for ``big'' groups, questions of uniqueness and regularity are simplified. The method is then applied to prove that for a constellation of explicit manifolds $ M$, the cone $ C(M) = \{ tx;x \in M$ and $ 0 \leqslant t \leqslant 1\} $ is the unique solution to the Plateau problem for $ M$, (Thus, there is no hope for general interior regularity of solutions in codimension one.) These manifolds include the original examples of type $ {S^n} \times {S^n} \subset {{\text{R}}^{2n + 2}},n \geqslant 3$, due to Bombieri, DeGiorgi, Giusti and Simons. They also include a new example in $ {{\text{R}}^8}$ and examples in $ {{\text{R}}^n}$ for $ n \geqslant 10$ with any prescribed Betti number nonzero.

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Keywords: Plateau problem, integral current, $ G$-invariant current, minimal cone, interior regularity
Article copyright: © Copyright 1972 American Mathematical Society

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