Defines horocyclic evolutes, parallels, and involutes of spacelike frontals in hyperbolic 2-space.
problem None explicitly stated; focuses on definitions and relations.
method Using enveloid theorem, defines horocyclic parallel and involute as normal envelopes of horocycles.
result Investigates relations among horocyclic evolutes, parallels, and involutes.
New surfaces with special geodesic and horocycle behaviors discovered.
problem Understanding geodesic and horocycle dynamics on hyperbolic surfaces.
method Constructing geometrically infinite hyperbolic surfaces with tailored recurrence properties.
result First examples of non-trivial minimal horocyclic orbit closures and infinite locally-finite conservative horocyclic invariant measures.
Non-ergodic measures found in horocycle flow on Abelian differentials.
problem Finding non-ergodic measures in the horocycle flow on Abelian differentials.
method Analyzing weak convergence of ergodic measures to non-ergodic invariant measures.
result Existence of points with non-equidistributing horocycle flow orbits.
A bijection preserving horocycles/hypercycles is an isometry in hyperbolic plane.
problem Understanding transformations of constant curvature curves in hyperbolic geometry.
method Analyzing bijections that map horocycles to horocycles and hypercycles to hypercycles.
result Every abstract automorphism of geodesic/horocycles/hypercycles graphs is induced by an earthquake map/isometry.
Study horocycle orbits in Z-covers of hyperbolic surfaces.
problem Classify horocycle orbit closures in Z-covers of compact hyperbolic surfaces. method Careful analysis of distance minimizing geodesic rays in the cover.
result All non-maximal horocycle orbit closures have integer Hausdorff dimension.
Example of divergent horocycle in Teichmüller space.
problem Characterizing convergence in Teichmüller spaces.
method Constructing a specific curve and horocycle in a punctured sphere, then generalizing.
result Found a divergent horocycle in Teichmüller spaces of complex dimension greater than one.
Study geometric actions of groups on horocyclic products.
problem Understanding geometric actions of groups on horocyclic products.
method Analyzing geometric actions of groups on horocyclic products of CAT(-κ) spaces.
result Groups acting on horocyclic products are either ascending HNN extensions of finitely-generated virtually nilpotent groups or not finitely presented.
Classifies horocycle flow closures in hyperbolic 3-manifolds.
problem Classifying horocycle flow closures in hyperbolic 3-manifolds.
method Classifies orbit closures of the 1-dimensional horocycle flow on the frame bundle of M.
result The closure of a horocycle in M is a properly immersed submanifold.
The paper proves inequalities for hyperbolic sets and curves.
problem Proving inequalities for sets and curves in hyperbolic geometry.
method Defining horocyclic Minkowski sums and proving inequalities for hyperbolic areas.
result Horocyclic Brunn-Minkowski inequality holds for hyperbolic sets.
Pack hyperbolic surfaces with circles or horocycles, noting symmetries.
problem Pack hyperbolic surfaces efficiently.
method Discuss packing with circles or horocycles, noting symmetries.
result Observations on symmetries of packed hyperbolic surfaces.
A support theorem for the horocycle Radon transform f \to \hat{f} is a property of the form \hat{f} of compact support \rightarrow f of compact support. Here we prove a variation of this result where support (\hat{f}) is outside a fixed horocycle in hyperbolic space.
New surfaces show horocyclic flow isn't always minimal.
problem Complex dynamics on infinite fineness surfaces.
method Construction of infinite hyperbolic surfaces.
result Horocyclic flow is not minimal on infinite fineness surfaces.
We introduce a deformation of Riemann surfaces and we are interested in the convergence of this deformation to a point of the Gardiner-masur boundary of Teichmueller space. This deformation, which we call the horocyclic deformation, is directed by a projective measured foliation and belongs to a certain horocycle in a …
Using zippered rectangle coordinates we parametrize a Poincaré section for horocycle flow on the space of genus 2 translation surfaces with one singular cone point of angle 6π. In addition, we bound the return time under horocycle flow to this Poincaré section by examining a subset of surfaces where a certain sum of …
Let M be a closed oriented surface endowed with a Riemannian metric g and let Ω be a 2-form. We show that the magnetic flow of the pair (g,Ω) has zero asymptotic Maslov index and zero Liouville action if and only g has constant Gaussian curvature, Ω is a constant multiple of the area form of g and the mag…
In this work we investigate the following isoperimetric problem in the hyperbolic plane: to find the regions of prescribed area with minimal perimeter between two parallel horocycles. We give an explicit and detailed description of all such regions.
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…
We consider flows, called Wu flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of Wu flows and we show that Wu flows have purely absolutely continuous spectrum in the orthocom…
The paper defines catenary curves in spheres and hyperbolic planes.
problem Defining catenary curves in non-Euclidean geometries.
method Characterizations of catenary curves in terms of curvature and angle with geodesics.
result Characterizations and extensions of catenary curves in hyperbolic geometry.
Study CMC hypersurfaces in H2imesH2 with a specific symmetry.
problem Constant mean curvature hypersurfaces with double horocyclic symmetry in H2imesH2. method Reduction to a single ODE, solving explicitly, classifying solutions.
result Existence and uniqueness of double horocyclic CMC hypersurfaces.
Proof of Graustein's theorem in different geometries.
problem Average curvature of plane ovals and convex curves in various geometries.
method Wave propagation approach for different geometries.
result The average curvature is attained at least at four points in different geometries.
The horocyclic flow on geometrically infinite surfaces shows recurrent irregular orbits or non-minimal closures.
problem Complex dynamics of horocyclic flow on geometrically infinite surfaces.
method Analyzing the recurrence and minimality of irregular orbits.
result Irregular orbits are recurrent or have non-hR minimal closures. We construct a Poincaré section for the horocycle flow on the modular surface SL(2,R)/SL(2,Z), and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…
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.
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…
Study shows orbits on a specific surface without intersecting geodesics.
problem Understanding orbits on a specific surface without intersecting geodesics.
method Analyzing the horocyclic flow on the unit tangent bundle of an untwisted flute.
result Recurrent and irregular orbits do not intersect closed geodesics.
Proves effective slope gaps for lattice surfaces.
problem Proving effective slope gaps for lattice surfaces.
method Proves effective slope gap distribution for square torus and general lattice surfaces.
result Effective slope gap distribution result for lattice surfaces.
The study connects lamination and orbit closures in hyperbolic manifolds.
problem Understanding the geometric and dynamical properties of horocycle orbit closures in Z-covers of compact hyperbolic manifolds. method Exposes connections between distance minimizing laminations and horospherical orbit closures in Z-covers of compact hyperbolic manifolds. Provides novel constructions and explicit descriptions. result Even slight perturbations to hyperbolic metrics can drastically change horocycle orbit closures.
Introduces hyperbolic generalized framed surfaces and their properties.
problem None explicitly stated; focuses on introducing new geometric objects.
method Generalization of hyperbolic framed surfaces and curves.
result Established conditions for a surface to be a hyperbolic generalized framed base surface and explored their singularities.
Study saddle connections on hyperelliptic surfaces, finding growth rates.
problem Count saddle connections on hyperelliptic surfaces without interior intersections.
method Used horocycle renormalization to prove lower bound growth rate.
result Found saddle connections satisfy L(logL)d−2 growth rate. In this thesis we consider a way to construct a rich family of compact Riemann Surfaces in a combinatorial way. Given a 3-regualr graph with orientation, we construct a finite-area hyperbolic Riemann surface by gluing triangles according to the combinatorics of the graph. We then compactify this surface by adding finit…
A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such n-gons are parametrized by the subspaces of (0,∞)n that contain their side length collections,…
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.
Classifies involutions on spherical 3-manifolds.
problem Classifying involutions on spherical 3-manifolds.
method Geometric approach to conjugacy classification.
result Insights into topological properties of involutions.
Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.
problem Understanding non-Gorenstein involutions on Calabi-Yau threefolds.
method Classification of Calabi-Yau threefolds with specific properties.
result Classification of Calabi-Yau threefolds with Picard rank one and non-Gorenstein involutions.
The paper studies circular evolutes and involutes of framed curves in Euclidean space.
problem Investigating properties of framed curves and their evolutes and involutes.
method Definition and analysis of circular evolutes and involutes of framed curves, properties of normal surfaces, and their relations.
result Circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similar to fronts in the Euclidean plane.
New formula for dual knots using involutions.
problem Understanding dual knots and their transformations.
method Involutive analog of knot surgery formula.
result Computed local equivalence class for involutive dual knots.
The study classifies involutions on del Pezzo surfaces.
problem Classifying involutions on del Pezzo surfaces.
method Mapping class group theory and hyperbolic reflection groups.
result A complete classification of involutions on del Pezzo surfaces.
Characterizes the Legendre involution on generic frontals.
problem Identifying the Legendre involution on a specific class of frontals.
method Analyzes generic frontals under mild assumptions and uses complexification.
result Any involution with the same fixed points as the Legendre involution is the Legendre involution.
The paper defines conditions for good involutions in generalized Alexander quandles.
problem Determining conditions for good involutions in generalized Alexander quandles.
method Analyzing the structure of generalized Alexander quandles and their involutions.
result Classification of all good involutions in connected generalized Alexander quandles.
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
The paper develops a new theory for knots and 3-manifolds with involutions.
problem Developing a new theory for knots and 3-manifolds with involutions.
method Establishing a version of Seiberg-Witten Floer K-theory for knots and 3-manifolds with involutions.
result 10/8-type inequalities for knots and involutions, yielding lower bounds on stabilizing numbers and relative genera.
The study proves symplectic quandles cannot have good involutions.
problem Existence of good involutions in symplectic quandles.
method Investigation of necessary and sufficient conditions for good involutions.
result Nonexistence of good involutions in symplectic quandles.
Involutions generate mapping class groups of infinite surfaces.
problem Generating involutions for mapping class groups of infinite surfaces.
method Analyzing infinite surfaces with n ends, showing involutions generate groups for n ≥ 6 and n ≥ 3.
result Involutions generate mapping class groups for n ≥ 6 and n ≥ 3.
Minimal involutions generate a subgroup of nonorientable surfaces.
problem Generating a minimal set of involutions for a specific subgroup.
method Obtained a minimal generating set of involutions.
result Minimal involutions for the level 2 subgroup of a nonorientable surface.
Study exact surgery formula in involutive Heegaard Floer homology.
problem Understanding integer homology spheres through knot surgery.
method Using doubling model of involution and mapping cone formula.
result Examples of non-homology cobordant integer homology spheres.
We study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all n⩾3 PU(n,1) has involution length at most 8.
Anti-symplectic involutions connect a sphere in a symplectic surface.
problem Understanding involutions on Lagrangian spheres in symplectic quadrics.
method Using Hamiltonian isotopy to show connections between involutions.
result Anti-symplectic involutions are Hamiltonian isotopic.