On the free boundary variational problem for minimal disks. (Q2711845)

The author proves an existence theorem for the problem of extremizing the energy (equivalently, area) for maps from the unit disk \(D\) into a Riemannian manifold \(N^n\) having a boundary lying on a specified embedded submanifold \(M^m.\) Under the assumption that the relative homotopy group \(\pi_k (N, M)\not = 0\;(k\geq 2),\) the theorem states that the solutions exist and are of index \(\leq k-2.\) This paper also studies the second variation theory for the free boundary problem for minimal disks, and gives an instability theorem for the case where \(N\) is the unit ball in \(\mathbb R^n.\) Two interesting theorems in this paper are:NEWLINENEWLINEThe existence theorem. (Theorem 1 in the introduction) \ {Let \(N\) be a complete and homogeneously regular Riemannian manifold and \(M\subset N\) a compact submanifold. If the relative homotopy group \(\pi_k(N,M) \not = 0 \) where \(k\geq 2,\) then eitherNEWLINENEWLINE(i) there exists a nonconstant minimal disk in \(N\) meeting \(M\) orthogonally along \(\partial D\) of index \(\leq k-2,\) orNEWLINENEWLINE(ii) there exists a nonconstant two-sphere in \(N\) of index \(\leq k-2.\)}NEWLINENEWLINEThe instability theorem. (Theorem 2.9 in section 2.4) \ {Let \(\Sigma^l\) (of any dimension) be a minimal submanifold in the unit ball \(B\) in \(\mathbb R^n,\) with \(\partial\Sigma\subset \partial B,\) that meets \(\partial B\) orthogonally; then \(\Sigma\) is unstable.}NEWLINENEWLINEThe paper consists of two parts: Part 1 is the proof of the main existence theorem and contains five sections. Section 1.1 describes the perturbed problem; section 1.2 gives the main estimates that are needed for convergence; in section 1.3, the author produces critical maps for the \(\alpha\)-energy by applying Ljusternik-Schnirelman theory; section 1.4 and 1.5 describe the convergence of these maps as \(\alpha\rightarrow 1.\)NEWLINENEWLINEPart 2 contains four sections. Section 2.1 gives a complex version of the second variation of area; the proof of the main index estimates is in section 2.2 and 2.3; and the last section states the instability theorem with its proof.