On the Morse index of minimal surfaces in \({\mathbb{R}}^ p\) with polygonal boundaries
Let \(\Gamma \subset {\mathbb{R}}^ p\) (p\(\geq 3)\) denote a Jordan polygon with \(N+3\) (N\(\geq 1)\) vertices. Then the minimal surfaces x spanning \(\Gamma\) correspond to the critical points \(\overset\circ\tau\) of an analytic function \(\theta\) : \(T\to {\mathbb{R}}\) in N variables. This function was introduced by M. Shiffman and the regularity of \(\theta\) has been investigated by E. Heinz. Now we are interested in the correspondence of the second order: The second derivative of \(\theta\) is given by its Hessian M(\(\overset\circ\tau\)), the second variation of the minimal surface x is described by the Schwarz operator \(S_ x\). We first study the eigenvalue problem of this singular differential operator by variational methods. Counting the negative eigenvalues of \(S_ x\) we obtain the Morse index \(m_ x\) of x. As central result we prove: The Morse index \(m_ x\) of the minimal surface x coincides with the Morse index m(\(\overset\circ\tau\)) of \(\theta\) (the number of negative eigenvalues of M(\(\overset\circ \tau\))) at the corresponding critical point \(\overset\circ\tau\in T\). This is achieved by comparison of the finite and infinite dimensional eigenvalue problems of M(\(\overset\circ\tau\)) and \(S_ x\), using certain simultaneous variations of the minimal surface.
scientific article; zbMATH DE number 3981977
scientific article