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 be a compact submanifold of Euclidean space which is invariant by a compact group . When , it is shown that there always exists a solution to the Plateau problem for which is invariant by and, furthermore, that uniqueness of this solution among -invariant currents implies uniqueness in general. This result motivates the subsequent study of the Plateau problem for within the class of -invariant integral currents. It is shown that this equivariant problem reduces to the study of a corresponding Plateau problem in the orbit space 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 , the cone and is the unique solution to the Plateau problem for , (Thus, there is no hope for general interior regularity of solutions in codimension one.) These manifolds include the original examples of type , due to Bombieri, DeGiorgi, Giusti and Simons. They also include a new example in and examples in for with any prescribed Betti number nonzero.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1972-0308905-4

Keywords:
Plateau problem,
integral current,
-invariant current,
minimal cone,
interior regularity

Article copyright:
© Copyright 1972
American Mathematical Society