Embedded minimal surfaces in manifolds diffeomorphic to the three-dimensional ball or sphere
By the famous theorem of Lusternik-Schnirelman there exist at least three closed geodesics without selfintersections on \((S^ 2,g)\) where g is an arbitrary Riemannian metric. Here, the author establishes analogues of this result for minimal surfaces. Specifically, he is able to prove the following beautiful theorems. Theorem 1. Let A be a compact body in \({\mathbb{R}}^ 3\), homeomorphic to the closed unit ball, with boundary \(\partial A\) of class \(C^ 4\), and of positive mean curvature with respect to the interior normal. Then A contains at least three embedded minimal disks meeting \(\partial A\) orthogonally along their boundary, i.e. solving a free boundary problem with boundary \(\partial A\). Theorem 2. Let M be a compact three- dimensional Riemannian manifold, diffeomorphic to the three-sphere \(S^ 3\). Then M contains at least four embedded minimal two-spheres. The numbers three resp. four are what one would expect from the results of Lusternik and Schnirelman. As in the author's previous paper [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 401-426 (1986; Zbl 0669.49024)] and in the joint paper with the reviewer [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 345-390 (1986; Zbl 0617.49017)] the minimal surfaces are obtained by a minimax-procedure. This saddle-point construction in the context of geometric measure theory was first introduced by Pitts. Here, one has to use higher order saddle-point arguments to get more than one minimal surface. However, to exclude multiple coverings one has to construct new classes in which new saddle- point constructions are then performed. For a number of technical details the reader is referred to the author's previous paper mentioned above.
scientific article; zbMATH DE number 4117402
scientific article