Embedded Plateau problem
The Plateau problem asks for the existence of an area minimizing disk for a given simple closed curve in a manifold. After this problem was solved for \(3\)-dimensional Euclidean space by Douglas and Rado, it was generalized to Riemannian manifolds including a regularity of solutions by many people. However, these area minimizing disks may not be embedded, even though the curves bound an embedded disk in the ambient manifold. In decades, this question of embeddedness of the area minimizing disk has been studied.NEWLINENEWLINEIn this paper, the author considers the same embeddedness question. But instead of considering the question ``for which curves must the area minimizing disks be embedded?'', he analyzes the structure of the surface which minimizes area among the embedded disks whose boundary is any given simple closed curve. He proves that if \(\Gamma\) is a simple closed curve bounding an embedded disk in a closed \(3\)-manifold \(M\), then there exists a disk \(\Sigma\) with \(\partial \Sigma = \Gamma\) such that \(\Sigma\) minimizes the area among all the embedded disks bounding \(\Gamma\). Moreover, \(\Sigma\) is minimal and smoothly embedded everywhere except where the boundary \(\Gamma\) meets the interior of \(\Sigma\).
scientific article; zbMATH DE number 6024126
scientific article