Degree theory on oriented infinite-dimensional varieties and the Morse number of minimal surfaces spanning a curve in \({\mathbb{R}}^ n\)
The author introduces for almost all closed curves \(\Gamma\) in \({\mathbb{R}}^ n\) (n\(\geq 3)\) an invariant, called the ''Morse number''. In a certain sense, this invariant counts the number of minimal surfaces spanning \(\Gamma\). The author shows that this invariant is equal to 1 and as a consequence, he proves a strong version of the mountain-pass lemma of Morse-Tompkins. The construction of the Morse number and the proof rely on degree theory for Fredholm vector fields on Banach manifolds. For dimension \(n=3\), an important difficulty has to be overcome: due to the index theorem of Böhme and the author, almost all wires \(\Gamma\) in \({\mathbb{R}}^ n\) bound finitely many nondegenerate minimal surfaces for \(n\geq 4\), while for \(n=3\) this is not true.
scientific article; zbMATH DE number 4009364
scientific article