We prove an optimal systolic inequality for nonpositively curved Dyck's surfaces. The extremal surface is flat with eight conical singularities, six of angle theta and two of angle 9pi - theta, for a suitable theta with cos(theta) in Q(sqrt{ 19}). Relying on some delicate capacity estimates, we also show that the extremal surface is not conformally equivalent to the hyperbolic surface with maximal systole, yielding a first example of systolic extremality with this behavior.
Categories: math.DG math.GT math.MG