Compact non-orientable surfaces of genus 6 with extremal metric discs

Abstract

A compact hyperbolic surface of genus g is said to be extremal if it admits an extremal disc, a disc of the largest radius determined only by g. We discuss how many extremal discs are embedded in non-orientable extremal surfaces of genus 6. This is the final genus in our interest because it is already known for g = 3, 4, 5, or g > 6. We show that non-orientable extremal surfaces of genus 6 admit at most two extremal discs. The locus of extremal discs is also obtained for each surface. Consequently non-orientable extremal surfaces of arbitrary genus g 3 admit at most two extremal discs. Furthermore we determine the groups of automorphisms of non-orientable extremal surfaces of genus 6 with two extremal discs.

A brute force computer aided proof of an existence result about extremal hyperbolic surfaces

“…Notice that four of the eleven required surfaces can be found already in the literature, since they correspond to the cases for which k N = 1 (see [GN07], [Nak09], [Nak12], [Nak13] and [Nak16]). These surfaces correspond to N = 12, 18, 24 or 30.…”

Extremal k-packings in compact non-orientable surfaces

“…He also proved that the inequality becomes an equality precisely when X is uniformized by a torsion-free Fuchsian group K acting in the unit disc D such that at some point z the Dirichlet domain D(K, z) := {w ∈ D : d(w, z) ≤ d(w, γ(z)), ∀γ ∈ K} is a (hyperbolic) regular polygon with angle 2π/3. The corresponding surfaces are known as extremal surfaces, and have been widely investigated in the literature both in the orientable and in the nonorientable case (see [Bav96], [GGD99], [BV02], [GN07], [Nak16]) .…”

“…For example, for k = 1 Theorem 8 says that 1-packings in compact non-orientable surfaces will be unique if 6g − 6 is not in the above list of numbers, which precisely occurs for g > 6 (consistent with [GN07]). In the series of papers [GN07], [Nak09], [Nak12], [Nak13] and [Nak16] all 1-extremal surfaces of genus g from 3 to 6 were studied in detail, and in particular the existence of extremal surfaces with more than one extremal disc was explicitly shown for these values of g. Now, suppose that there exists a compact non-orientable primitive k N -extremal surface X of genus g N with multiple extremal k N -packings. Denote N = 6g N + 6k N − 12 k .…”

“…Four of the eleven surfaces we need have occurred already in the literature, namely the ones corresponding to the cases for which k N = 1 (see [GN07], [Nak09], [Nak12], [Nak13] and [Nak16]). These cases correspond to N = 12, 18, 24 or 30.…”