The index theorem for \(k\)-fold connected minimal surfaces
Section 0 is an introduction to Plateau's problem. Section 1 describes the analytical properties of conformality operators. Section 2 shows that: (1) the set of minimal \(k\)-fold connected immersions in \(R^ n\) of branching type \(\lambda\) whose boundary curve belongs to \({\mathcal A}=\{\alpha: \partial \Omega_{{\bar \rho}}\to R^ n|\) \(\alpha\) immersion of class \(H^ r\}\) can be viewed as a differentiable submanifold of \({\mathcal A}\times {\mathcal D}\), where \({\mathcal D}=\cup_{\rho \in P}{\mathcal D}_{\rho}\), \({\mathcal D}_{\rho}=\{u: \partial \Omega_{\rho}\to \partial \Omega_{{\bar \rho}}|\) u diffeomorphism of class \(H^ s\) with deg u\(=1\) on each component of \(\partial \Omega_{\rho}\}\); (2) at a minimal immersion, the conformality operator restricted to the manifold of surfaces of branching type \(\lambda\) is a submersion. Section 3 gives geometric applications.
scientific article; zbMATH DE number 3899791