Dyck's surfaces, systoles, and capacities

TL;DR

This paper establishes an optimal systolic inequality for nonpositively curved Dyck's surfaces, identifying the extremal surface with specific conical singularities and demonstrating it is not conformally equivalent to the hyperbolic surface with maximal systole.

Contribution

It provides the first example of systolic extremality that differs conformally from the hyperbolic maximal systole surface, using capacity estimates.

Findings

Optimal systolic inequality for Dyck's surfaces.

Extremal surface has specific conical singularities.

Extremal surface is not conformally hyperbolic with maximal systole.

Abstract

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.