Non-uniqueness of obstacle problem on finite Riemann surface (Q2494625)

Let \(S\) be a Riemann surface which is constructed from a compact one by removing finitely many points. Let \(\varphi\) be a quadratic differential on this surface. One can define both a norm \(| | \varphi| | \) of \(\varphi\) and the \(\varphi\)-height of a closed curve \(\gamma\). One can, by considering an infinum, construct a hight of a free homotopy class of curves. The author calls a compact subset \(E\) of \(S\) an \textit{obstacle} if \(S \setminus E\) is connected and if \(E\) is contained in a topological disc in \(S\). One now considers the Teichmüller class of \(S \setminus E\) which respects punctures. One considers the points in the Teichmüller space where the norm of the image of \(\varphi\) is maximal; the image is defined through the heights. Under certain conditions on \(S,E,\varphi\) the author shows that the extremal Riemann surfaces are not unique. This is based on an inequality due to F. P. Gardiner. The author also discusses an illustrative example of this phenomenon.