Assume both X and Y are Riemann surfaces which are subsets of compact Riemann surfaces X1 and Y1, respectively, and that the set X1 −X has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on X and Y are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows...

We study the behavior of the Cheeger isoperimetric constant on infinite families of graphs and Riemann surfaces, and its relationship to the first eigenvalue λ1 of the Laplacian. We adapt probabilistic arguments of Bollobás to the setting of Riemann surfaces, and then show that Cheeger constants of the modular surfaces are uniformly bounded from above away from the maximum value. We extend this...

An explicit canonical construction of monopole connections on non trivial U(1) bundles over Riemann surfaces of any genus is given. The class of monopole solutions depend on the conformal class of the given Riemann surface and a set of integer weights. The reduction of Seiberg-Witten 4-monopole equations to Riemann surfaces is performed. It is shown then that the monopole connections constructe...

We prove a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, showing that in general an N=2 superconformal (resp. N=1 superanalytic) DeWitt superRiemann surface is N=2 superconformally (resp., N=1 superanalytically) equivalent to a manifold with transition functions containing no odd functions of the even variable if and only if the first...

Belyi’s Theorem [Be] of 1979 had a profound effect on Galois Theory, Riemann surfaces and complex algebraic curves. It led Grothendieck [Gr] to develop his theory of dessins d’enfants in which there has been a great interest. The theory is about embedding graphs into compact Riemann surfaces. For further applications to topics such as such as moduli spaces and Physics the reader is recommended ...

We show that a family of line bundles of degree zero over a plumbing family of Riemann surfaces with a separating (resp. non-separating) node p admits a nice (resp. almost nice) family of flat p-singular Hermitian metrics. As a consequence, we give necessary and sufficient conditions for a family of line bundles over such families of Riemann surfaces to admit an (almost) nice family of p-singul...

Let g be an integer ≥ 3 and let Bg = {X ∈ Mg |Aut(X) = 1d}, where Mg denotes the moduli space of compact Riemann surfaces of genus g. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space, we prove that the subloci corresponding to Riemann surfaces with automorphism groups isomorphic to cyclic groups of order 2 a...