Gou Nakamura

Compact non-orientable hyperbolic surfaces with an extremal metric disc

The size of a metric disc embedded in a compact non-orientable hyperbolic surface is bounded by some constant depending only on the genus g ≥ 3. We show that a surface of genus greater than six contains at most one metric disc of the largest radius. For the case g = 3, we carry out an exhaustive study of all the extremal surfaces, finding the location of every extremal disc inside them.

Compact non-orientable surfaces of genus 4 with extremal metric discs

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.

Parametrizations of some Teichmüller spaces by trace functions

We show a tuple of trace functions which give a global parametrization of the Teichmüller space T (g,m) of types (1, 2) and (2, 0). We also show that the mapping class group acting on these Teichmüller spaces can be represented by a group of rational transformations in seven variables.

Compact Klein surfaces of genus 5 with a unique extremal disc

A compact (orientable or non-orientable) surface of genus g is said to be extremal if it contains an extremal disc, that is, a disc of the largest radius determined only by g. The present paper concerns non-orientable extremal surfaces of genus 5. We represent the surfaces as side-pairing patterns of a hyperbolic regular 24-gon, that is, a generic fundamental region of an NEC group uniformizing each of the surfaces. We also describe the group of automorphisms of the surfaces with a unique extremal disc.