Related papers: Triangulations of branched affine surfaces

The moduli space of (1,7)-polarized abelian surfaces with a level structure was shown by Manolache and Schreyer to be rational with compactification a Fano 3-fold of genus 12 which is also a compactification of the variety of powersum…

In a previous paper, we showed how certain orientations of the edges of a graph G embedded in a closed oriented surface S can be understood as discrete spin structures on S. We then used this correspondence to give a geometric proof of the…

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results…

We classify compact surfaces with torsion-free affine connections for which every geodesic is a simple closed curve. In the process, we obtain completely new proofs of all the major results concerning the Riemannian case. In contrast to…

We consider translation surfaces with poles on surfaces. We shall prove that any finite group appears as the automorphism group of some translation surface with poles. As a direct consequence we obtain the existence of structures achieving…

In this article, we focus on a very special class of foliations with complex leaves whose diffeomorphism type is fixed. They have a unique compact leaf and the noncompact leaves all accumulate onto it. We show that the complex structure…

We show that the description of the holomorphic $\mathbb C \mathrm P^1$-bundle associated to a holomorphic projective structure on a Riemann surface in terms of the principal bundle of projective $2$-frames extends very well to the setting…

We show that an arbitrary infinite graph can be compactified by its ${\aleph_0}$-tangles in much the same way as the ends of a locally finite graph compactify it in its Freudenthal compactification. In general, the ends then appear as a…

We show that the tangle space of a graph, which compactifies it, is a quotient of its Stone-\v{C}ech remainder obtained by contracting the connected components.

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…

The triangulations of a surface $\Sigma$ with a prescribed set of vertices can be endowed with a graph structure $\mathcal{F}(\Sigma)$. Its edges connect two triangulations that differ by a single arc. It is known that, when $\Sigma$ is a…

We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by…

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a…

An affine surface is said to be an affine Zoll surface if all affine geodesics close smoothly. It is said to be an affine almost Zoll surface if thru any point, every affine geodesic but one closes smoothly (the exceptional geodesic is said…

For any subgroup of $\mathrm{SL}(3,\mathbb{R})\ltimes\mathbb{R}^3$ obtained by adding a translation part to a subgroup of $\mathrm{SL}(3,\mathbb{R})$ which is the fundamental group of a finite-volume convex projective surface, we first show…

We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a…

We study the cohomological classification of vector bundles on smooth real affine surfaces and threefolds. We show that, as was observed in joint work in A. Asok and J. Fasel and in a coming joint paper with S. Banerjee and J. Fasel, under…

The fundamental theorem of affine geometry says that a self-bijection $f$ of a finite-dimensional affine space over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then $f$ of the expected…

Affine rotation surfaces are a generalization of the well-known surfaces of revolution. Affine rotation surfaces arise naturally within the framework of affine differential geometry, a field started by Blaschke in the first decades of the…

Smooth manifolds have been always understood intuitively as spaces with an affine geometry on the infinitesimal scale. In Synthetic Differential Geometry this can be made precise by showing that a smooth manifold carries a natural structure…