Related papers: The $(2,5)$ minimal model on genus two surfaces
Consider the moduli space $\mathcal{M}_{g}$ of Riemann surfaces of genus $g\geq 2$ and its Deligne-Munford compactification $\bar{\mathcal{M}_{g}}$. We are interested in the branch locus ${\mathcal{B}_{g}}$ for $g>2$, i.e., the subset of…
The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form…
We extend the planar Pfaffian formalism for the evaluation of the Ising partition function to lattices of high topological genus g. The 3D Ising model on a cubic lattice, where g is proportional to the number of sites, is discussed in…
Let $X$ be a compact Riemann surface of genus $g$ and let $x \in X$. We derive the classical presentation of $\pi_1(X,x)$ (i.e the one given by $2g$ generators $a_1,b_1, \dots, a_g,b_g$ and the relation $\prod_{i=1}^g[a_i,b_i] = 1$) from…
This article deals with the construction of surfaces that are suitable for representing pentachords or 5-pitch segments that are in the same $T/I$ class. It is a generalization of the well known \"Ottingen-Riemann torus for triads of…
For compact Riemann surfaces, the collar theorem and Bers' partition theorem are major tools for working with simple closed geodesics. The main goal of this paper is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic…
We introduce the notion of log-Riemann surfaces. These are Riemann surfaces given by cutting and pasting planes together isometrically, and come equipped with a holomorphic local diffeomorphism to C called the projection map, and a…
Inspired by the Taubes-Wu construction of $\mathcal{C}^{1,\alpha}$ two-valued harmonic functions by the use of symmetry, we construct minimal surfaces with stratified branching sets as graphs of $\mathcal{C}^{1,\alpha}$ two-valued…
We provide quantitative evidence for our previous conjecture which states an equivalence of the partition function of a 3d N=2 gauge theory on a duality wall and that of the SL(2,R) Chern-Simons theory on a mapping torus, for a class of…
We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises…
We view all smooth metrics $g$ on a closed surface $\Sigma$ through their Nash isometric embeddings $f_g: (\Sigma,g) \rightarrow (\mathbb{S}^{\tilde{n}}, \tilde{g})$ into a standard sphere of large, but fixed, dimension $\tilde{n}$. We…
We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form $P(\Omega)T^q(\Omega)|\Omega|^{-2q-1/2}$ and the class of…
We propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for…
We study and classify regular and semi-regular tessellations of Riemann surfaces of various genera and investigate their corresponding supersymmetric gauge theories. These tessellations are generalizations of brane tilings, or bipartite…
We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two…
We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $\Gamma\subset \mathbb R^4$ such that the unique area minimizing surface spanned by $\Gamma$ has infinite…
We prove the existence of complete minimal surfaces of genus g>1 which minimize the total curvature for their genus. Our method is first to identify this (Weierstrass high dimensional period) problem with the problem of finding a particular…
Starting from a Riemannian conformal structure on a manifold M, we provide a method to construct a family of Lorentzian manifolds. The construction relies on the choice of a metric in the conformal class and a smooth 1-parameter family of…
In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class. In this…
We study minimal complex surfaces S of general type with q(S)=q and p_g(S)=2q-3, q>= 5. We give a complete classification in case that S has a fibration onto a curve of genus >=2. For these surfaces K^2=8\chi. In general we prove that…