Geometry on surfaces, a source for mathematical developments
Résumé
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results for such structures. Higher-dimensional analogues are also discussed. Some constructions with Riemann surfaces lead, by analogy, to notions that hold for arbitrary fields, and not only the field of complex numbers. The Riemann sphere is also defined using surjective homomorphisms of real algebras from the ring of real univariate polynomials to (arbitrary) fields, in which the field with one element is interpreted as the point at infinity of the Gaussian plane of complex numbers. Several models of the hyperbolic plane and hyperbolic 3-space appear, defined in terms of complex structures on surfaces, and in particular also a rather elementary construction of the hyperbolic plane using
real monic univariate polynomials of degree two without real roots. Several notions and problems connected with conformal structures in dimension 2 are discussed, including dessins d'enfants, the combinatorial characterization of polynomials and rational maps of the sphere, the type problem, uniformization, quasiconformal mappings, Thurston's characterization of Speiser graphs, stratifications of spaces of monic polynomials, and others. Classical methods and new techniques complement each other.
The final version of this paper will appear as a chapter in the Volume Surveys in Geometry. II (ed. A. Papadopoulos), Springer Nature Switzerland, 2024.
Mots clés
geometric structure
conformal structure
almost complex structure ($J$-field)
Riemann sphere
uniformization
the type problem
rigidity
model for hyperbolic space
cross ratio
Belyi's theorem
Riemann--Hurwitz formula
Chasles 3-point function
branched covering
type problem
dessin d'enfants
slalom polynomial
slalom curve
space of monic polynomials
stratification
fibered link
divide
Speiser curve
Speiser graph
line complex
quasiconformal map
almost analytic function.
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