Related papers: Real Riemann Surfaces: Smooth and Discrete

Let $Mod_{g}$ be the modular group of surfaces of genus $g$. Each element $[h]\in Mod_{g}$ induces in the integer homology of a surface of genus $g$ a symplectic automorphism $H([h])$ and Poincar\'{e} shown that $H:Mod_{g}\to…

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface.…

The moduli space ${\mathcal{M}}_{g}$, of genus $g\geq2$ closed Riemann surfaces, is a complex orbifold of dimension $3(g-1)$ which carries a natural real structure i.e. it admits an anti-holomorphic involution $\sigma$. The involution…

Classification of real K3 surfaces X with a non-symplectic involution \tau is considered. For some exactly defined and one of the weakest possible type of degeneration (giving the very reach discriminant), we show that the connected…

In this informal expository note, we quickly introduce and survey compactifications of strata of holomorphic 1-forms on Riemann surfaces, i.e. spaces of translation surfaces. In the last decade, several of these have been constructed,…

This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic…

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given…

Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In…

This paper studies timelike minimal surfaces in the De Sitter space $\mathbb S^3_1(1) \subset \mathbb R^4_1$ via a complex variable. Using complex analysis and stereographic projection of lightlike vectors we obtain a representation…

In this paper we develop a general approach to dimer models analogous to Krichever's scheme in the theory of integrable systems. We start with a Riemann surface and the simplest generic meromorphic functions on it and demonstrate how to…

In the context of complex algebraic varieties, the decomposition theorem for semi-small maps provides a decomposition of the direct image of the constant sheaf. In this work, we develop a decomposition theorem for branched coverings of…

The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces.

The Hodge decomposition is a fundamental result in differential geometry and algebraic topology, particularly in the study of differential forms on a Riemannian manifold. Despite extensive research in the past few decades,…

We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is…

In this paper we develop an intrinsic formalism to study the topology, smooth structure, and Riemannian geometry of the Wasserstein space of a closed Riemannian manifold. Our formalism allows for a new characterisation of the Weak topology…

We consider a log-Riemann surface $\mathcal{S}$ with a finite number of ramification points and finitely generated fundamental group. The log-Riemann surface is equipped with a local holomorphic difffeomorphism $\pi : \mathcal{S} \to \C$.…

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Fr\"olicher spaces, another generalisation of smooth structures. We then…

We consider a compact Riemann surface $\mathscr{R}$ with a complex of non-intersecting Jordan curves, whose complement is a pair of Riemann surfaces with boundary, each of which may be possibly disconnected. We investigate conformally…

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disc removed. We define a refined Teichmueller space of such Riemann surfaces and demonstrate that in…

We use discrete holomorphic polynomials to prove that, given a refining sequence of critical maps of a Riemann surface, any holomorphic function can be approximated by a converging sequence of discrete holomorphic functions.