A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Study shows genus two Heegaard diagrams for all Thurston geometries.
Article explores Thurston's circle packing theorem in 3-manifold geometry.
Survey on Thurston metric on Teichmüller space, focusing on extremal maps.
The paper extends Menelaus' and Ceva's theorems to translation triangles in various Thurston geometries.
Study of Teichmüller space geometry using infinitesimal and global methods.
Historical notes on Thurston's 3-manifold geometry.
Explains Thurston geometries and visualization techniques.
Survey on four-dimensional Thurston geometries with Riemannian metrics.
The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.
Overview of Thurston's work in math.
Ray-marching method visualizes 8 Thurston geometries in real-time.
Classifies geodesic-preserving bijections in Thurston geometries.
Proof of Thurston's earthquake theorem using Anti-de Sitter geometry.
This is an expository paper. We prove the Cannon-Thurston property for bounded geometry surface groups with or without punctures. We prove three theorems, due to Cannon-Thurston, Minsky and Bowditch. The proofs are culled out of earlier work of the author.
We introduce the notion of manifolds of amalgamation geometry and its generalization, split geometry. We show that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map.
In three dimensions, a `master theory' for all Thurston geometries requires imaginary flux. However, these geometries can be obtained from physical three-dimensional theories with various additional scalar fields, which can be interpreted as moduli in various compactifications of a higher-dimensional `master theory'. T…
We find analogues of the Willmore functional for each of the Thurston geometries with 4-dimensional isometry group such that the CMC-spheres in these geometries are critical points of these functionals.
3-manifolds explained through geometry, proving Thurston's conjecture.
We construct new explicit proper biharmonic functions on the -dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $H^2\times\rn$ and $S^2\times\rn$.
Classifies homogeneous hypersurfaces in specific 4D geometries.
The paper classifies hypersurfaces in a specific 4D geometry.
Survey of combination theorems in geometry and dynamics.
The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map. This is an exposition of a special case of th…
For any positive natural number we construct new explicit proper -harmonic functions on the celebrated -dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $\H^2\times\rn$ and $\s^2\times\rn$.
The paper shows how profinite completions can reveal 4D geometries except for specific cases.
We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface . Some of our results on the coarse geometry of this metric apply to arbitrary surfaces of finite type; however, we focus particular attention on the case where the surface is a once-punct…
We obtain an ordering of closed aspherical 4-manifolds that carry a non-hyperbolic Thurston geometry. As application, we derive that the Kodaira dimension of geometric 4-manifolds is monotone with respect to the existence of maps of non-zero degree.
Geometrization says `` any closed oriented three-manifold which is prime (not a connected sum) carries one of the eight Thurston geometries OR it has incompressible torus walls whose complementary components each carry one of four particular Thurston geometries" (see Introduction and Figure 1). These geometric componen…
The paper classifies Ricci solitons and studies harmonic vector fields on a specific Thurston geometry.
Study of harmonic Riemannian submersions from 3D geometries.
Study of hypersurfaces in Sol4_0 geometry, classifying parallel and totally umbilical types.
We extend Thurston's metric to projective filling currents, embedding Teichmüller space into the larger space.
Geometrization Theorem solves complex geometry problems.
Let N^h be a hyperbolic 3-manifold of bounded geometry corresponding to a hyperbolic structure on a pared manifold (M,P). Further, suppose that (\partial{M} - P) is incompressible, i.e. the boundary of M is incompressible away from cusps. Further, suppose that M_{gf} is a geometrically finite hyperbolic structure on (M…
As an example of the transitions between some of the eight geometries of Thurston, investigated before, we study the geometries supported by the cone-manifolds obtained by surgery on the trefoil knot with singular set the core of the surgery. The geometric structures are explicitly constructed. The most interesting phe…
The seven non euclidean geometries of the Thurston's geometrization program are proved to originate naturally from singularization morphisms and versal deformations on euclidean 3-manifolds generated in the frame of the Langlands global program. The Poincare conjecture for a 3-manifold appears as a particular case of t…
This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces,…
Defines and classifies Thurston geometries and connects simplicial volume to Kodaira dimension.
We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curve…
In this note, we provide a description of the structure of homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jorgensen-Thurston Theorem in hyperbolic geometry.
LCD n-manifolds are linked to branched n-manifolds.
About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed manifold, Mostow rigidity assures that a hyperbolic structure is unique when it ex…
Abstract notes on hyperbolic surfaces and Teichmüller spaces.
We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhof…
Study shows diffeomorphism groups of certain 3-manifolds retract to isometry groups.
Algorithm computes Thurston norm for hyperbolic 3-manifolds.
This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2 \times R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie group…
Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further s…