Related papers: The grafting map of Teichmuller space

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disc removed. We define a refined Teichmueller space of such Riemann surfaces and demonstrate that in…

We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…

Due to its geometric properties, hyperbolic space can support high-fidelity embeddings of tree- and graph-structured data, upon which various hyperbolic networks have been developed. Existing hyperbolic networks encode geometric priors not…

In the space $\mathcal{H}^2$ of hyperbolic surfaces decorated with a base unit vector, the topology induced by the Gromov-Hausdorff convergence coincides with the Chabauty topology on the space of discrete torsion-free subgroups of…

We extend the notion of graph homomorphism to cellularly embedded graphs (maps) by designing operations on vertices and edges that respect the surface topology; we thus obtain the first definition of map homomorphism that preserves both the…

Generalising an example by Girondo and Wolfart, we use finite group theory to construct Riemann surfaces admitting two or more regular dessins (i.e. orientably regular hypermaps) with automorphism groups of the same order, and in many cases…

We cut a hyperbolic surface of finite area along some analytic simple closed curves, and glue in cylinders of varying moduli. We prove that as the moduli of the glued cylinders go to infinity, the Fenchel-Nielsen twist coordinates for the…

This work studies certain aspects of graphs embedded on surfaces. Initially, a colored graph model for a map of a graph on a surface is developed. Then, a concept analogous to (and extending) planar graph is introduced in the same spirit as…

Modern methods of graph theory describe a graph up to isomorphism, which makes it difficult to create mathematical models for visualizing graph drawings on a plane. The topological drawing of the planar part of a graph allows representing…

We explicitly construct a symplectomorphism that relates magnetic twists to the invariant hyperk\"ahler structure of the tangent bundle of a Hermitian symmetric space. This symplectomorphism reveals foliations by (pseudo-) holomorphic…

We investigate the problem of finding smooth hypersurfaces of constant mean curvature in hyperbolic space, which can be represented as radial graphs over a subdomain of the upper hemisphere. Our approach is variational and our main results…

The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichm\"uller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow…

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective scheme and is dominated…

Tilings of the hyperbolic plane are of significant interest among many branches of mathematics, physics and computer science. Yet, their construction remains a non-trivial task. Current approaches primarily use tree-based recursive…

We consider principal fibre bundles with a given connection and construct almost complex structures on the total space if the adjoint bundle is isomorphic to the tangent bundle of the base. We derive the integrability condition. If the…

We use the reflection group trick to glue manifolds with corners that are Borel-Serre compactifications of locally symmetric spaces of noncompact type and obtain aspherical manifolds. We call these \emph{piecewise locally symmetric}…

Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…

The theme of this survey is that subgroups of the mapping class group of a finite type surface S can be studied via the geometric/dynamical properties of their action on the Thurston compactification of the Teichmuller space of S, just as…

We prove a quantitative version of the non-uniform hyperbolicity of the Teichm\"uller geodesic flow. Namely, at each point of any Teichm\"uller flow line, we bound the infinitesimal spectral gap for variations of the Hodge norm along the…

If a graph is in bridge position in a 3-manifold so that the graph complement is irreducible and boundary irreducible, we generalize a result of Bachman and Schleimer to prove that the complexity of a surface properly embedded in the…