Related papers: Conformal mapping of rectangular heptagons
We apply the technique of jet differentials to establish a Gauss curvature estimate for an open Riemann surface $M$, equipped with a conformal metric induced from a nonconstant holomorphic map that is highly ramified over a generic…
In this article, we summarize the results on symmetric conformal geometries. We review the results following from the general theory of symmetric parabolic geometries and prove several new results for symmetric conformal geometries. In…
For all open Riemann surface M and real number $\theta \in (0,\pi/4),$ we construct a conformal minimal immersion $X=(X_1,X_2,X_3):M \to \mathbb{R}^3$ such that $X_3+\tan(\theta) |X_1|:M \to \mathbb{R}$ is positive and proper. Furthermore,…
Consider discrete conformal maps defined on the basis of two conformally equivalent triangle meshes, that is edge lengths are related by scale factors associated to the vertices. Given a smooth conformal map $f$, we show that it can be…
We present a new and simple proof of Teichm\"uller-Wittich-Belinskii's and Gutlyanskii-Martio's theorems on the conformality of quasiconformal mappings at a given point. Known proofs gave separate estimates for the radial and angular…
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…
The aim of this work is an analytic investigation of differential equations producing mirror maps as well as giving new examples of mirror maps; one of these examples is related to (rational approximations to) $\zeta(4)$. We also indicate…
Branched covering Riemann surfaces $(\mathbb{C},f)$ are studied, where $f$ is the Euler Gamma function and the Riemann Zeta function. For both of them fundamental domains are found and the group of covering transformations is revealed. In…
We review some of our recent work on the conformal geometry corresponding to the triangulated surfaces used in 2-dimensional simplicial quantum gravity. In particular, we discuss the regularized Liouville action associated with random Regge…
We show that the space of theta functions on tropical tori is identified with a convex polyhedron. We also show a Riemann-Roch inequality for tropical abelian surfaces by calculating the self-intersection numbers of divisors.
In this paper, we give a description of the possible poles of the local zeta function attached to a complex or real analytic mapping in terms of a log-principalization of an ideal associated to the mapping. When the mapping is a…
In Part I, we develop the notions of a Moebius structure and a conformal Cartan geometry, establish an equivalence between them; we use them in Part II to study submanifolds of conformal manifolds in arbitrary dimension and codimension. We…
This is an elementary review of our recent work on the classification of the spectra of those two-dimensional rational conformal field theories (RCFTs) whose (maximal) chiral algebras are current algebras. We classified all possible…
Conformal mapping may be the best-known topic in complex analysis. Any simply connected nonempty domain $\Omega$ in the complex plane ${{\mathbb{C}}}$ (assuming $\Omega\ne {{\mathbb{C}}}$) can be mapped bijectively to the unit disk by an…
In this short letter, we reformulate the Riemann zeta function using the holographic framework of the celestial conformal field theory. For spacetime dimension larger than our Minkowski spacetime $M^4$, the Riemann zeta function is…
In this note we demonstrate how the analogy between the harmonic Gauss map of a constant mean curvature surface and the harmonic conformal Gauss map of a Willmore surface can be used to obtain results on Willmore surfaces.
We present a computational approach to general hyperelliptic Riemann surfaces in Weierstrass normal form. The surface is either given by a list of the branch points, the coefficients of the defining polynomial or a system of cuts for the…
The concept of an adapted homology basis for a prime order conformal automorphism of a compact Riemann surface extends to arbitrary finite groups of conformal automorphisms. Here we compute some examples of adapted homology bases for some…
We define a set of holomorphic functions in terms of the Hauptmodul of a quotient Riemann surface and prove that these functions are holomorphic on the upper half-plane. It is also shown that these functions are automorphic forms of weight…
Introduction to two dimensional conformal field theory on open and unoriented surfaces. The construction is illustrated in detail on the example of SU(2) WZW models.