Systols of singular plane surfaces of genus two
Given a closed non-simply-connected two-manifold \(M\) with a metric \(g\), we denote by \(\text{sys}(M,g)\) the length of the shortest noncontractible closed geodesic and call \(\sigma(M,g) = \frac{\text{area}(M)}{\text{sys}(M,g)}\) the systolic constant of the metric. A metric is called extremal if it is a local minimum of the functional \(\sigma(M,g)\). The infimum of \(\sigma(M,g)\) taken over all metrics is positive. Calabi conjectured that certain flat metrics with conical singularities on oriented surfaces of genus \(2\) and \(3\) are extremal. The author proves that no flat metric with conical singularities on an oriented closed surface of genus two is extremal.
scientific article; zbMATH DE number 2133502