Related papers: The boundary-Wecken classification of surfaces

In this paper we continue the study started in part I (posted). We consider a planar, bounded, $m$-connected region $\Omega$, and let $\bord\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\bord\Omega$,…

In this article, we prove that if $(M,g)$ is a genus $G$ orientable surface with a single boundary component $S^1$, and if $(D,g_0)$ is a disc such that interior points are connected by unique geodesics and $$d_{(D,g_0)}(x,y) \geq…

Let $\partial{\bf H}^n_{\mathbb K}$ denote the boundary of a symmetric space of rank-one and of non-compact type and let $d_{\mathfrak{H}}$ be the Kor\'anyi metric defined in $\partial{\bf H}^n_{\mathbb K}$. We prove that if $d$ is a metric…

In this paper we study normal surfaces whose anticanonical divisors are strictly nef, i.e. (-K)C>0 for every curve C.

We consider two natural classes of minimal laminations in three-manifolds. Both classes may be thought of as limits - in different senses - of embedded minimal disks. In both cases, we prove that, under a natural geometric assumption on the…

In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary…

In a paper from 2011, Jiang, Wang and Zhang studied the fixed points and fixed subgroups of selfmaps on a connected finite graph or a connected compact hyperbolic surface $X$. In particular, for any selfmap $f: X\to X$, they proved that a…

In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles…

Given a nonconstant holomorphic map f: X -> Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor R_f, which describes the locus at which f fails to be locally injective. The divisor…

We investigate the behavior of non-Euclidean plates with constant negative Gaussian curvature using the F\"oppl-von K\'arm\'an reduced theory of elasticity. Motivated by recent experimental results, we focus on annuli with a periodic…

Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the covering group $\pi_1(X)$ on $\tilde{X}$ is of the first kind-i.e., the surface $X$ is equal to its convex core. We first prove…

In a paper from 1960, Felix Browder established a theorem concerning the continuation of the fixed points of a family of continuous functions $f_t:X\to X$ depending continuously on a parameter $t\in [0,1]$, where $X$ is a convex and compact…

In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by…

Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichm\"{u}ller space of the boundary. This paper examines the skinning map of a 3-manifold M, a genus-2 handlebody with two rank-1 cusps.…

This article is concerned with locally flatly immersed surfaces in simply-connected $4$-manifolds where the complement of the surface has fundamental group $\mathbb{Z}$. Once the genus and number of double points are fixed, we classify such…

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $X\times\mathbb{A}^n\cong X'\times\mathbb{A}^n$ for (affine) algebraic varieties $X$ and $X'$ implies that $X\cong X'$. In this paper we provide a…

We prove that every bounded type Siegel disk of a rational map must be a quasi-disk with at least one critical point on its boundary. This verifies Douady-Sullivan conjecture for bounded type Siegel disks.

Numerical Campedelli surfaces are minimal surfaces of general type with p_g=0 (and so q=0) and K^2=2. Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical…

A vector space partition of $\mathbb{F}_q^v$ is a collection of subspaces such that every non-zero vector is contained in a unique element. We improve a lower bound of Heden, in a subcase, on the number of elements of the smallest occurring…

The $p$-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace--Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the…