Interior Regularity of Solutions to the Isotropically Constrained Plateau Problem

In this paper, we study the regularity of isotropically areaminimizing surfaces. We prove a partial regularity theorem which says that if an W^1,2 isotropic map from a two-dimensional disk into ℝ²ⁿ minimizes area relative to its boundary among isotropic competitors and is close enough in W^1,2 norm to a linear holomorphic isotropic map, then it is smooth in the interior. Furthermore, we prove that the solution to the isotropically constrained Plateau problem exists and has a smooth interior with possibly isolated singularities.

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Published 1 January 2003