Gabino González-Diez

On Beauville surfaces

We prove that if a finite group G acts freely on a product of two curves C1×C2 so that the quotient S = C1×C2/G is a Beauville surface then C1 and C2 are both non hyperelliptic curves of genus ≥ 6; the lowest bound being achieved when C1 = C2 is the Fermat curve of genus 6 and G = (Z/5Z). We also determine the possible values of the genera of C1 and C2 when G equals S5, PSL2(F7) or any abelian group. Finally, we produce examples of Beauville surfaces in which G is a p-group with p = 2, 3.

Genus two extremal surfaces: Extremal discs, isometries and Weierstrass points

It is known that the largest disc that a compact hyperbolic surface of genusg may contain has radiusR=cosh−1(1/2sin(π/(12g−6))). It is also known that the number of such (extremal) surfaces, although finite, grows exponentially withg. Elsewhere the authors have shown that for genusg>3 extremal surfaces contain only one extremal disc.Here we describe in full detail the situation in genus 2. Following results that go back to Fricke and Klein we first show that there are exactly nine different extremal surfaces. Then we proceed to locate the various extremal discs that each of these surfaces possesses as well as their set of Weierstrass points and group of isometries.

On extremal discs inside compact hyperbolic surfaces

Abstract It has been proved by C. Bavard that the radius of a disc isometrically embedded in a compact hyperbolic surface of genus g is bounded by R g = cosh ⁡ − 1 ( 1 2 sin ⁡ π 12 g - 6 ) , and that surfaces containing discs of such extremal radius are found in every genus. By constructing explicit surfaces of genus 2, he has also shown that extremal discs may or may not be unique. Here we show that a compact surface of genus g > 3 has at most one embedded extremal disc.

Beauville Surfaces with Abelian Beauville Group

A Beauville surface is a rigid surface of general type arising as a quotient of a product of curves

Belyi's theorem for complex surfaces

C_{1}

ON THE NUMBER OF COINCIDENCES OF MORPHISMS BETWEEN CLOSED RIEMANN SURFACES

,

On a conjecture of Whittaker concerning uniformization of hyperelliptic curves

C_{2}

On extremal Riemann surfaces and their uniformizing fuchsian groups

of genera

Conformal Versus Topological Conjugacy of Automorphisms on Compact Riemann Surfaces

g_{1},g_{2}\ge 2

The arithmeticity of a Kodaira fibration is determined by its universal cover

by the free action of a finite group