arXiv subjects

Ernesto Girondo

Publications and source records attributed to Ernesto Girondo.

5 recordsLinked to original sources

arXiv2026

Dirichlet domains and the NEC uniformization of extremal hyperbolic surfaces

Extremal hyperbolic surfaces are finite-type hyperbolic surfaces that contain an embedded disc of the largest possible radius. We introduce a new general approach to these objects, based on the uniformization by NEC groups, obtaining a structural description of the Dirichlet domains associated to extremal disc centers for surfaces with cusps and/or geodesic boundary. We also provide a number of applications of our new description of extremal surfaces, as an effective counting formula for extremal disc configurations, the determination of every such configuration with Euler characteristic -1 and -2, the location of hidden extremal discs in several families of extremal surfaces, and the full description of their automorphism groups.

math.GT↗

arXiv2023

The minima of the geodesic length functions of uniform filling curves

There is a natural link between (multi-)curves that fill up a closed oriented surface and dessins d'enfants. We use this approach to exhibit explicitly the minima of the geodesic length function of a kind of curves (uniform filling curves) which include those that admit a homotopy equivalent representative such that all self-intersection points as well as all faces of their complement have the same multiplicity. We show that these minima are attained at the Grothendieck-Belyi surfaces determined by a natural dessin d'enfant associated to these filling curves. In particular they are all Riemann surfaces defined over number fields.

math.GT↗

arXiv2019

Extremal k-packings in compact non-orientable surfaces

An extremal $k$-packing is a collection of $k$ mutually disjoint metric discs, embedded in a surface, whose radius is maximal for the given topology. We study compact non-orientable surfaces of genus $g\ge 3$ containing extremal $k$-packings.

math.MG↗

arXiv2019

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

Extremal compact hyperbolic surfaces contain a packing of discs of the largest possible radius permitted by the topology of the surface. It is well known that arithmetic conditions on the uniformizing group are necessary for the existence of a second extremal packing in the same surface, but constructing explicit examples of this phenomenon is a complicated task. We present a brute force computational procedure that can be used to produce examples in all cases.

math.MG↗

arXiv2009

Right triangles with algebraic sides and elliptic curves over number fields

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field Q(λ) (depending on n) and an explicit point P_λof infinite order in the Mordell-Weil group of the elliptic curve Y^2=X^3-n^2*X over Q(λ).

math.NT↗