Harmonic maps between flat surfaces with conical singularities
This paper deals with the local and global mapping behavior of energy minimizing maps from a surface \(\Sigma\) into a surface \(S\) which is flat except at a finite set \(\mathfrak S\) of singularities. It is required that near each \(q\in {\mathfrak S}\) the surface \(S\) looks like the cone over a curve of length \(2\pi\alpha\) with \(\alpha>1\) so that each of the vertices contributes to negative curvature. The author firstly proves the existence of minimizers in a homotopy class using the Bochner formula in an approximation argument, and classifies the homogeneous harmonic maps into three different types, one of which is the typical singularity of a Teichmüller map. Then, some regularity results for local minimizers are obtained through blow up arguments. These may be improved to a complete description of the mapping behavior of minimizers in a homotopy class of degree one. A global condition of this type is given naturally because the main problem is to analyse the preimage of the vertices under the map. The author shows that the minimizer is unique and that the preimage of a cone point is either an isolated point or a finite union of analytic arcs meeting at certain angle. It is proved that the second case occurs always except when the cone metric is the Teichmüller metric on the target. Finally, the author proves that the Teichmüller map minimizes energy in its homotopy class with respect to that metric.
scientific article; zbMATH DE number 922486