Proof of Thurston's earthquake theorem using Anti-de Sitter geometry.
problem Proving Thurston's earthquake theorem for orientation-preserving homeomorphisms.
method Using the bi-invariant geometry of Anti-de Sitter three-space.
result Provided a proof of Thurston's earthquake theorem.
The paper generalizes Thurston's earthquake map to cluster algebras of finite type.
problem Tackling Thurston's earthquake map in the context of cluster algebras of finite type.
method Introducing a cluster algebraic generalization of Thurston's earthquake map, defined by gluing exponential maps.
result Proves an analogue of the earthquake theorem for cluster algebras of finite type, showing the cluster earthquake map is a homeomorphism.
Let $\cT$ be Teichmüller space of a closed surface of genus at least 2. For any point $c\in \cT$, we describe an action of the circle on $\cT\times \cT$, which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of $\cT$. This circle action shares some of the …
The earthquake flow is asymmetric and cannot be extended to an SL(2,R) action.
problem The asymmetry of Thurston's earthquake flow and its implications.
method Analysis of orbifold automorphisms and measured geodesic laminations.
result The earthquake flow does not extend to an SL(2,R) action and lacks continuous self-symmetries.
Study on earthquake metric on Teichmüller space, proving properties and new completions.
problem Understanding the earthquake metric on Teichmüller space.
method Proofs of properties, new completions, and interpretation of the metric.
result Coincidence of various completions for the earthquake metric.
Paper defines and studies measures related to earthquakes and best Lipschitz maps.
problem Understanding Thurston's conjecture about maps and measures on hyperbolic surfaces.
method Examining Lie algebra valued transverse measures and their relation to earthquakes.
result Defines and shows correspondence between best Lipschitz maps and earthquakes.
The Teichmuller unipotent flow can be defined concretely on certain moduli spaces of singular flat surfaces by shearing polygonal presentations of the surfaces. Thurston's earthquake flow on moduli spaces of hyperbolic surfaces is more mysterious. Both flows have deep and important connections to other areas of mathema…
This paper adapts Thurston's earthquake metric to Riemann surfaces with marked points.
problem Defining a norm and metric on Teichmüller spaces for surfaces of arbitrary genus.
method Adapting Thurston's earthquake norm to Riemann surfaces with marked points and using complex Legendre transforms.
result Establishes a complete analogue of Thurston's earthquake norm in the conformal setting.
A measured laminations on the universal hyperbolic solenoid § is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal hyperbolic solenoid § is uniquely determined by a measured lamination on §; it is a leafwise earthquake with…
Extends earthquake and horocycle flows to new measures.
problem Ergodic theory of earthquake flow on measured laminations.
method Generalizes shear coordinates to arbitrary measured laminations.
result Classifies ergodic measures for P action on bundle of quadratic differentials.
We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into th…
Method computes centers of Poisson and skein algebras for loops on surfaces.
problem Computing centers of Poisson and skein algebras associated to loops on surfaces.
method Systematic method using Goldman and Wolpert's Poisson algebras and Turaev's skein algebras.
result Computed centers of various Poisson and skein algebras for finite type hyperbolic surfaces.
The paper proves a theorem about earthquake extensions of vector fields on circles.
problem Proving a theorem about earthquake extensions of vector fields on circles.
method Using the geometry of the dual of Minkowski three-space and Half-pipe three-geometry.
result A generalization of Kerckhoff's and Gardiner's infinitesimal earthquake theorems to a broader setting.
Geodesic envelopes stay uniformly bounded in specific Teichmüller spaces.
problem Understanding the behavior of geodesics in Teichmüller spaces.
method Identifying extremal geodesics, computing Fenchel-Nielsen twisting, and estimating earthquake path lengths.
result Width of geodesic envelopes is uniformly bounded in specific Teichmüller spaces.
New geometric definition of Lie bracket for undirected curves.
problem Understanding the Lie bracket of undirected curves on a surface.
method Local geometric definition and proof of three results.
result The TWG bracket counts intersection and suggests disjoint representatives.
For a compact surface X0, Thurston introduced a compactification of its Teichmüller space T(X0) by completing it with a boundary PML(X0) consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichmüller space T(X0) of a noncompact Ri…
This paper was first written in 1990, but was never published. In it, the author presents a novel approach to the study of constant curvature spacetimes in 2+1 dimensions. A parameterization of flat 2+1-dimensional domains of dependence is given in terms of measured geodesic laminations. There is also an interesting re…
We prove an "Earthquake Theorem" for hyperbolic metrics with geodesic boundary on a compact surfaces S with boundary: given two hyperbolic metrics with geodesic boundary on a surface with k boundary components, there are 2k right earthquakes transforming the first in the second. An alternative formulation arises…
Unique geodesics selected by energy minimization in Teichmüller space.
problem Finding a unique geodesic between points in Teichmüller space.
method Energy minimization of harmonic map rays, extending Thurston boundary.
result Selection of a unique Thurston geodesic through points in Teichmüller space.
We prove two related results. The first is an ``Earthquake Theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than π: any two such metrics in are connected by a unique left earthquake. The second result is that the space of ``globally hyperbolic'' AdS manifolds with ``parti…
Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.
problem Classifying mapping class groups and rational maps.
method Unified proof following Bers' approach.
result Unified proof of Nielsen-Thurston classification.
The paper extends Thurston's method to new variants of Mather-Thurston theorem.
problem Proving new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
method Generalizing Thurston's technique to prove new variants of Mather-Thurston theorem for PL homeomorphisms and contactomorphisms.
result The paper answers questions posed by Gelfand-Fuks and Greenberg on PL foliations and Rybicki on contactomorphisms.
Article explores Thurston's circle packing theorem in 3-manifold geometry.
problem Understanding Thurston's circle packing theorem in 3-manifold geometry.
method Analyzes the Koebe-Andre'ev-Thurston Theorem and its relation to Thurston's circle packing theorem.
result Illustrates the significance of Thurston's circle packing theorem in 3-manifold geometry.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.
The paper extends Menelaus' and Ceva's theorems to translation triangles in various Thurston geometries.
problem Extending classical theorems to non-Euclidean geometries.
method Using projective models of Thurston geometries and defining a ``surface of a translation-like triangle".
result Generalization of Menelaus' and Ceva's theorems to non-constant curvature Thurston geometries.
Study earthquake deformations on a once-punctured torus.
problem Understanding earthquake deformations on Teichmüller space.
method Two methods: linear recurrence relations and hyperbolic geometry.
result Algebraic and geometric interpretations of earthquake deformations.
Modernizes Thurston's proof of entropy theorem for traintrack maps.
problem Proving the entropy theorem for traintrack maps using Thurston's methods.
method Modernizes Thurston's original proof, fills gaps, and proves ergodicity.
result A cohesive proof of the traintrack theorem, including ergodicity.
Deep neural networks predict earthquake locations with high accuracy.
problem Predicting the location of earthquakes with high precision.
method Recurrent Convolutional Neural Networks (R-CNN) model that accounts for spatio-temporal dependencies.
result Neural networks model outperforms baseline models in predicting earthquakes with ROC AUC 0.975 and PR AUC 0.0890.
Study shows house buyers in Christchurch value earthquake risk differently based on time since 2011 quake.
problem Understanding how house buyers' perception of earthquake risk changes over time.
method Used a hedonic price model to analyze house prices in Christchurch over three periods.
result Buyers value earthquake risk differently based on the time since the 2011 Christchurch earthquake.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
In this paper we study the typical speed of a generic earthquake trajectory leaving compact sets in the moduli space of the once-punctured torus. Mirzakhani showed that the earthquake flow is measurably equivalent to the horocyclic flow, which has been studied extensively. Our main result shows that the earthquake flow…
Developed theory for Thurston maps with a small set of essential singularities.
problem Characterizing Thurston maps with essential singularities.
method Analyzed pullback maps on Teichmüller space to characterize Thurston maps.
result Established a characterization theorem for Thurston maps with four postsingular values.
Teichmüller space rigidity proven for Thurston metric.
problem Understanding isometries in Teichmüller space with Thurston metric.
method Analyzing R-linear surjective isometries between cotangent spaces. result Every isometry between hyperbolic surfaces induces an isometry in Teichmüller space.
Deviance Voronoi residuals improve earthquake insurance risk assessment.
problem Assessing earthquake insurance risk using spatio-temporal point process models.
method Extended Voronoi residuals and created simulation-based approach.
result Proposed formula for country-wide minimum capital test.
We prove that the bijective correspondence between the space of bounded measured laminations MLb(H) and the universal Teichmüller space T(H) given by λ↦Eλ∣S1 is a homeomorphism for the Fréchet topology on MLb(H) and the Teichmüller topology on T(H), where $E^λ…
EQShapelets detect earthquakes with high accuracy and interpretability.
problem Automated detection and cataloging of earthquakes.
method Time-series shape-based approach embedded in machine learning.
result EQShapelets detected all cataloged and 281 uncataloged events with lower false detection rate.
Study of infinity in Teichmüller space using Thurston boundary.
problem Metric structure on Thurston boundary of Teichmüller space.
method Developed tools in sup metrics and applied Minsky's theorem.
result Investigation of metric structure on Thurston boundary.
Two flexible, degenerate constructions related to Thurston's theorem.
problem Understanding the structure and local non-rigidity of Teichmüller spaces and their representations.
method Constructing geodesic segments and open sets in Teichmüller spaces with specific properties.
result Geodesic segments and open sets with degenerate properties in Teichmüller spaces.
We give a short proof of the fact that bounded earthquakes of the unit disk induce quasisymmetric maps of the unit circle. By a similar method, we show that symmetric maps are induced by bounded earthquakes with asymptotically trivial measures.
Continuity of earthquake flow map transfers Teichmüller dynamics results.
problem Transfer results from Teichmüller dynamics to earthquake flow.
method Analyze continuity of earthquake flow map and its inverse.
result Transfer results from Teichmüller dynamics to earthquake flow.
3-manifolds explained through geometry, proving Thurston's conjecture.
problem Thurston's conjecture about 3-manifolds.
method Explain geometrisation theorem and its proof by Perelman.
result Proved Thurston's conjecture about 3-manifolds.
Survey of recent Kleinian representation convergence results.
problem Kleinian representation convergence
method Survey and analysis of recent results following Thurston's theorems
result Survey of recent and less recent results on convergence of Kleinian representations
Historical notes on Thurston's 3-manifold geometry.
problem Exploring Thurston's notes on 3-manifold geometry.
method Historical commentary and references.
result Discussion of Lobachevsky, Andreev, and Milnor's works.
We establish a form of the h-principle for the existence of foliations quasi-complementary to a given one; the same methods also provide a proof of the classical Mather-Thurston theorem.
Survey of combination theorems in geometry and dynamics.
problem Combination theorems in hyperbolic geometry, group theory, and dynamics.
method Survey and focus on Thurston's contributions.
result Thurston's influence on combination theorems.
We prove a version the local Reeb-Thurston stability theorem for symplectic foliations.
A machine learning surrogate model predicts earthquake-induced building responses.
problem Expensive FE model simulations for earthquake damage estimation.
method SVD-based earthquake characterization and machine learning model training.
result Deep neural network provides most accurate predictions of building responses.
New benchmark for earthquake forecasting models shows current neural point processes are not yet suitable.
problem Lack of a modern benchmark for evaluating neural point process models in earthquake forecasting.
method Curated and standardized earthquake catalog, evaluation protocols, and datasets.
result None of the tested NPPs outperformed the classical ETAS model.