Related papers: Conformal mapping of rectangular heptagons

Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous…

For a function defined on an arbitrary subset of a Riemann surface, we give conditions which allow the function to be extended conformally. One folkloric consequence is that two common definitions of an analytic arc in ${\mathbb C}$ are…

We prove some differential equations for the Riemann theta function associated to the Jacobian of a Riemann surface. The proof is based on some variants of a formula by Fay for the theta function, which are motivated by their analogues in…

For a Riemann surface with cusps we define a theta function using the eigenvalues of the Laplacian and the singularities of the scattering determinant. We provide its meromorphic continuation and discuss its singularities.

This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using…

The aim of this note is to propose an interpretation for the full (non-chiral) correlation functions of the Liouville conformal field theory within the context of the quantization of spaces of Riemann surfaces.

We derive a representation formula for harmonic polynomials and Laurent polynomials in terms of densities of the double-layer potential on bounded piecewise smooth and simply connected domains. From this result, we obtain a method for the…

This paper solves the problem of computing conformal structures of general 2-manifolds represented as triangle meshes. We compute conformal structures in the following way: first compute homology bases from simplicial complex structures,…

Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of $(n-1)$-dimensional polytopes associated with two combinatorial families of rectangulations composed of $n$ rectangles.…

We derive a class of variational functionals which arise naturally in conformal geometry. In the special case when the Riemannian manifold is locally conformal flat, the functional coincides with the well studied functional which is the…

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations…

In this paper we make a detailed and self-contained study of the conformalGauss map. Then, starting from the seminal work of R. Bryant and the notion of conformal Gauss map, we recover many fundamental properties of Willmore surfaces. We…

We construct and fully characterize a scalar boundary conformal field theory on a triangulated Riemann surface. The results are analyzed from a string theory perspective as tools to deal with open/closed string dualities.

Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical…

In this note we give a simple relation between conformal mapping and the first eigenvalue of Laplacian for surfaces in Euclidean spaces.

We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a…

Surface mapping plays an important role in geometric processing. They induce both area and angular distortions. If the angular distortion is bounded, the mapping is called a {\it quasi-conformal} map. Many surface maps in our physical world…

We study the gluing theory of Riemann surfaces using formal algebraic geometry, and give computable relations between the associated parameters for different gluing processes. As its application to the Liouville conformal field theory, we…

We study counting functions of planar polygons arising from homological mirror symmetry of elliptic curves. We first analyze the signature and rationality of the quadratic forms corresponding to the signed areas of planar polygons. Then we…

The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the…