Perfect Riemann surfaces of genus 4 and 6 (Q2388654)

Let \(T_g\) be the Teichmüller space of the compact marked Riemannian surfaces of genus \(g\). The surface \(X\in T_g\) is said to be perfect if the gradients for the Weil-Petersson metric generates affinely the tangent space \(T_X(T_g)\). The systole of a Riemannian surface of genus \(g\) is defined as the minimum of geodesic length functions on the Teichmüller space. The surface realizing local maximum is called extremal one. The author constructs a new extremal surface and two perfect nonextremal surfaces of genus 4, the first examples of such surfaces of small genus. The idea of the method is to geometrically realize the groups of automorphisms with 4 points of ramification. The author obtains a new extremal surface of genus 6, and also an infinite sequence of perfect nonextremal surfaces of genus \(g\geq 4\). Besides, the proposed method permits to recover, in unified manner, the surfaces of genus \(g\leq5\), obtained by P.~Schmutz Schaller.