Birkhoff minimax principle for minimal surfaces with a free boundary
The author proves a multiplicity result on the number of nonconstant minimal surfaces with a free boundary on a convex hypersurface \(S\) in \(\mathbb{R}^{n+1}\). He shows that if \(S\) satisfies the following pinching condition: there exists a diffeomorphism \(f:(S^n,g_0) \to(S,g)\), with \(S^n\) unit sphere in \(\mathbb{R}^{n+1}\) and \(g_0\) the standard metric on \(S^n\), such that \(g_0\leq f^*(g)\leq 2g_0\) \((f^*(g)\) is the pull-back metric on \(S^n)\), then there exist at least \(l(n)\) distinct nonconstant minimal surfaces with a free boundary on \(S\), where \(l(n)= n+2^k-1\), \(2^k\leq n<2^{k+1}\) is the cuplength of the Grassmannian \(G(n+1,2)\). The proof is based on a perturbation method used by several authors to study harmonic maps between Riemannian manifolds. The perturbation method is needed since the Palais-Smale condition for the variational problem associated is not satisfied. The author develops also a Birkhoff min-max theory for perturbed functionals.
scientific article; zbMATH DE number 1337026