Harmonic maps with fixed singular sets
Let \(\Omega\) be a smooth domain in \(\mathbb{R}^ m\) and \(N\) a compact smooth Riemannian manifold. The authors study the space \({\mathcal H}\) consisting of harmonic maps \(u:\Omega\to N\) that have as singular set a fixed compact subset \(Z\) of \(\Omega\), having finite \((m-3)\)-dimensional Minkowski content. The topology on \({\mathcal H}\) is defined by using Hölder norms on derivatives weighted by powers of distances to \(Z\). The authors study the structure of \({\mathcal H}\) and other properties such as stability and minimality under perturbations. For smooth harmonic maps they prove that \({\mathcal H}\) is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite dimensional space of Jacobi fields with controlled singular behavior.
scientific article; zbMATH DE number 118433
scientific article