Short proof for ideal polygons with near optimal orthogeodesic decomposition.
problem Decomposing ideal polygons into orthogeodesics.
method Short proof with orthogeodesic decomposition of length at most 2log(n). result Optimal orthogeodesic decomposition of ideal polygons with length 2log(n). The paper explores orthogeodesics on hyperbolic surfaces and their integer traces.
problem Computing and understanding orthogeodesics on hyperbolic surfaces.
method Recursive method for computing orthogeodesic traces and combinatorial proof of Basmajian's identity.
result Existence of surfaces where orthogeodesic traces are integers.
Uniform counting formulas for orthogeodesics in Kleinian groups converge.
problem Counting orthogeodesics in Kleinian groups converging to a limit.
method Spectral gap of the limit manifold and geodesic flow mixing property.
result Asymptotically uniform counting formulas for orthogeodesics.
Proves identities linking curve lengths and orthogeodesics on hyperbolic surfaces.
problem Exploring relationships between curve lengths and orthogeodesics on hyperbolic surfaces.
method Partitioned orthogeodesics into sets based on dynamical behavior, relating them to geodesics on orbifold surfaces.
result Extends a result to surfaces with cusps, showing how to extend a previous result.
Explicit bounds found for shortest orthogeodesics and volumes of hyperbolic manifolds.
problem Finding explicit bounds for shortest orthogeodesics and volumes of hyperbolic manifolds.
method Derived explicit estimates for functions related to volumes and orthospectra, using a new approach.
result Explicit lower bound for the length of the shortest orthogeodesic in terms of volume.
This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.
problem Understanding relationships between geodesic and orthogeodesic lengths on hyperbolic surfaces.
method Investigates a broad family of identities involving lengths of all closed geodesics and orthogeodesics.
result Introduces new identities that include lengths of all closed geodesics, contrasting with previous identities.
The paper finds bounds on shortest dense curves on surfaces.
problem Finding shortest dense curves on surfaces.
method Quantitative density of closed geodesics and orthogeodesics.
result Upper bounds on shortest dense curves.
Graded identities for hyperbolic surfaces with cusps and cone points.
problem Understanding dilogarithm identities on hyperbolic surfaces.
method Establishing graded versions of Bridgeman's dilogarithm identity.
result Applications to the study of orthogeodesics.
For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian's identity from Teichmueller theory. Any equality characterizes diagonal embeddings.
Study geodesics on flat tori, focusing on convex bodies.
problem Analyze geodesics orthogonal to convex subsets on flat tori.
method Define anisotropic Sobolev spaces and study properties of geodesics.
result Compute residues of geometric Epstein function in terms of intrinsic volumes.
The Poincaré series for surfaces with boundary extends to the complex plane.
problem Counting geodesics on surfaces with boundaries.
method Analytic continuation of Poincaré series.
result Poincaré series extend meromorphically to the whole complex plane.
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.
Study on OI surfaces with unique geometric properties.
problem Characterizing and classifying ortho-integral surfaces.
method Analyzing geodesic arcs and cosh-length properties.
result Infinitely many commensurability classes of OI surfaces arise as topologies vary.
The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.
problem Counting arcs on hyperbolic surfaces with boundaries and cusps.
method Asymptotic analysis of pure mapping class group orbits and arc lengths.
result The number of arcs of bounded length is asymptotically proportional to L6g−6+2(n+p). The paper studies optimal maps between hyperbolic surfaces, focusing on their rigidity and obstructions.
problem Finding optimal Lipschitz maps between hyperbolic surfaces and understanding their rigidity and obstructions.
method Introducing deflations, optimal maps to trees that obstruct optimal maps between surfaces, and using a smooth orthogeodesic foliation.
result Deflations are the main obstructions to optimal maps between hyperbolic surfaces, and they are essentially the only ones.
Study geodesics on flat tori, focusing on orthogonal lengths and their distribution.
problem Properties of geodesics on flat tori and their lengths.
method Fine study of dynamical correlation function and anisotropic Sobolev spaces.
result Properties of the length distribution and singularities of its Fourier transform.
Geometrically boundary of surface moduli space defined.
problem Defining a geometric boundary for the moduli space of complex projective structures.
method Introducing a bordification of the moduli space PT(S) using projective classes of half-translation surfaces. result Established a homeomorphism between the moduli space and the product of Teichmüller and measured lamination spaces.