Embedded minimal annuli solving an exterior problem
Given a compact, strictly convex body \(X\) in \(\mathbb{R}^3\) and a closed Jordan curve \(\Gamma\) in \(\mathbb{R}^3 \setminus X\) the author is interested in area minimizing annuli which have \(\Gamma\) as one boundary component whereas the other one can move freely on \(\partial X\). For bodies of class \(C^{3, \alpha}\) and certain curves \(\Gamma\) (which are in some sense close to a planar circle) the author proves the existence of such minimizing annuli which are contained in the exterior of \(X\). This is a nontrivial result since the exterior of \(X\) is neither convex nor mean-convex. By applying methods of Meeks and Yau the author can prove the existence of such minimal annuli which are moreover embedded. By a limiting argument he obtains the existence of complete embedded minimal annuli with a free boundary on \(\partial X\) and a flat end at infinity. The crucial step in the existence proof consists in constructing a minimizing annulus whose free boundary on \(\partial X\) is not contractible on \(\partial X\setminus \{N, S\}\), where \(N\) and \(S\) denote the north- and southpole of \(\partial X\), respectively.
scientific article; zbMATH DE number 720495
scientific article