Embedded plateau problem

Coskunuzer, Baris

doi:10.1090/S0002-9947-2011-05486-3

Embedded plateau problem
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Trans. Amer. Math. Soc. 364 (2012), 1211-1224 Request permission

Abstract:

We show that if $\Gamma$ is a simple closed curve bounding an embedded disk in a closed $3$-manifold $M$, then there exists a disk $\Sigma$ in $M$ with boundary $\Gamma$ such that $\Sigma$ minimizes the area among the embedded disks with boundary $\Gamma$. Moreover, $\Sigma$ is smooth, minimal and embedded everywhere except where the boundary $\Gamma$ meets the interior of $\Sigma$. The same result is also valid for homogeneously regular manifolds with sufficiently convex boundary.

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