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We revisit Ahlfors theory of covering surfaces thanks to Stokes theorem.

In this paper, we will prove a result which is used by Guang-Yuan Zhang in another paper in which the existence of extremal surfaces for covering surfaces is proved and the sharp form of Ahlfors' Second Fundamental Theorem is given.

Ahlfors' theory of covering surfaces is one of the major mathematical achievement of last century. The most important part of his theory is the Second Fundamental Theorem (SFT). We are interested in the relation of errors of Ahlfors' SFT…

Ahlfors Second Fundamental Theorem of covering surfaces over the Riemann sphere are one of the major events in the history of function theory, which is a geometrical interpretation of the famous Nevanlinna's Second Fundamental Theorem in…

In the study of the constant in Ahlfors' second fundamental theorem involving a set E_{q} of q points, branch values of covering surfaces outside E_{q} bring a lot of troubles. To avoid this situation, for a given surface S, it is useful to…

We give two generalizations of the Clifford theorem to algebraic surfaces. As an application, we obtain some bounds for the number of moduli of surfaces of general type.

In this note we fill a gap in the proof of the main theorem (Theorem 1.2) of our paper 'Surfaces in 4-manifolds', Math. Res. Letters 4 (1997), 907-914.

In this paper continuing our work started in our earlier papers we prove the corona theorem for the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of finite type.

This is an elementary geometrical proof of Birkhoff theorem. It is hardly important, but the pictures behind are quite nice.

Covering theory is an important tool in representation theory of algebras, however, the results and the proofs are scattered in the literature. We give an introduction to covering theory at a level as elementary as possible.

The purpose of these notes is to present a fairly complete proof of the classification Theorem for compact surfaces. Other presentations are often quite informal (see the references in Chapter V) and we have tried to be more rigorous. Our…

We give an elementary and self-contained proof of the uniformization theorem for non-compact simply-connected Riemann surfaces.

A proof based on the Chern-Gauss-Bonnet Theorem is given to Hopf Theorem concerning the degree of the Gauss map of a hypersurface in $\mathbb{R}^n$.

We give a concise proof of the fundamental theorem of smoothing theory in the special case when a smoothing exists.

We prove a uniformization theorem in complex algebraic geometry.

We give a new simpler proof of a theorem of Jayne and Rogers.

We present a simple short proof of the Fundamental Theorem of Algebra, without complex analysis and with a minimal use of topology. It can be taught in a first year calculus class.

This paper contains a detailed, self contained and more streamlined proof of our $l^2$ decoupling theorem for hypersurfaces.

We give a proof of a phenomenon conjectured in our former article: "Beltrami forms, affine surfaces and the Schwarz-Christoffel formula: a worked out example of straightening". We also start an abstract discussion of the notion of limits of…

We prove an Ahlfors refection theorem for $p$-reflections over Jordan curves bounding subhyperbolic domains in $\hat {\mathbb C}$.