Related papers: The simple complexity of a Riemann surface

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable…

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the…

It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, including constructive proofs, but there has been little attention to the…

Let $\Sigma_g$ be a closed Riemann surface of genus $g$. Let $G$ be a finite subgroup of the automorphism group of $\Sigma_g$. It is well known that there exists a smooth $G$-equivariant embedding from $\Sigma_g$ to some Euclidean space…

We show that for a generic $8$-dimensional Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the curvature condition, we show that for a dense set of 8-dimensional Riemannian metrics…

This paper extends results of Hatcher and Vogtmann's work "Cerf Theory for Graphs" to ribbon graphs. Given an orientable, punctured and basepointed surface Sigma, we prove that the space of ribbon graphs that can be drawn in Sigma is…

We show that for every $\epsilon>0$, there exists a compact lamination by $\epsilon$-holomorphic surfaces in the complex projective plane, minimal, and that carries hyperbolic holonomy. We call $\epsilon$-holomorphic a real 2-dimensional…

Consider the strata of primitive $k$-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by $k$. The map that assigns to each $k$-differential the $k$-residues at these poles is a ramified…

Given a compact, oriented surface $S$ of finite genus and finitely many boundary components, we provide examples of finite covers $\tilde{S}$ of $S$ and non-simple closed curves $\gamma$ on $S$ which lifts to simple closed curves on…

In this paper, we show that if $R$ is a compact Riemann surface and $M=R\setminus\,\bigcup_i D_i$ is a domain in $R$ whose complement is a union of countably many pairwise disjoint smoothly bounded closed discs $D_i$, then $M$ is the…

Let $M$ be a compact 3--manifold with boundary a single torus. We present upper and lower complexity bounds for closed 3--manifolds obtained as even Dehn fillings of $M.$ As an application, we characterise some infinite families of even…

We give a topological interpretation of the core group invariant of a surface embedded in S^4. We show that the group is isomorphic to the free product of the fundamental group of the double branch cover of S^4 with the surface as a…

For compact regions Omega in R^3 with generic smooth boundary B, we consider geometric properties of Omega which lie midway between their topology and geometry and can be summarized by the term "geometric complexity". The "geometric…

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere…

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved K\"ahler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex…

A sequence of constant mean curvature surfaces $\Sigma_j$ with mean curvature $H_j \to \infty$ in a three-dimensional manifold $M$ condenses to a compact and connected graph $\Gamma$ consisting of a finite union of curves if $\Sigma_j$ is…

Consider genus $g$ curves that admit degree $d$ covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family $Y$ that naturally maps into the moduli space of…

We show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains…

Let $S$ be a nonorientable surface of genus $g\ge 5$ with $n\ge 0$ punctures, and $\Mcg(S)$ its mapping class group. We define the complexity of $S$ to be the maximum rank of a free abelian subgroup of $\Mcg(S)$. Suppose that $S_1$ and…

This note describes minimal surfaces $S$ of general type satisfying $p_g\geq 5$ and $K^2=2p_g$. For $p_g\geq 8$ the canonical map of such surfaces is generically finite of degree 2 and the bulk of the paper is a complete characterization of…