The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.
problem Finding the asymptotic behavior of shortest closed multi-geodesics on hyperbolic surfaces.
method Analyzing the length of shortest filling closed multi-geodesics using hyperbolic geometry and asymptotic analysis.
result The length of a shortest filling closed multi-geodesic is uniformly comparable to a specific formula involving the genus and lengths of closed geodesics.
Counting hyperbolic multi-geodesics with individual component lengths.
problem Counting hyperbolic multi-geodesics with specific component lengths.
method Unified geometric and topological techniques, combining Mirzakhani's results and Margulis's ideas.
result Asymptotic polynomial counts of multi-geodesics in mapping class group orbits, generalizing Wolpert's conjecture.
Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.
problem Distribution of shapes of complementary subsurfaces in moduli space.
method Study of shapes of complementary subsurfaces in moduli space as boundary lengths go to infinity.
result Random subsurfaces look like random ribbon graphs.
Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two different metrics on a surface have no geodesic arcs in common, but in special ca…
The study of random surfaces reveals asymptotic lengths of separating geodesics.
problem Understanding geometric properties of random hyperbolic surfaces.
method Analysis of Weil-Petersson measure and asymptotic behavior of lengths.
result The shortest separating closed geodesics have lengths about 2logg. Geometric bounds for low Steklov eigenvalues on hyperbolic surfaces with boundaries.
problem Finding lower bounds for low Steklov eigenvalues of hyperbolic surfaces with geodesic boundaries.
method Analysis of eigenfunction behavior on an adapted thick-thin decomposition for hyperbolic surfaces with geodesic boundaries.
result Sharp geometric lower bounds for low Steklov eigenvalues that depend on the shortest multi-geodesic disconnecting the surfaces.
Given a closed Riemannian manifold and a pair of multi-curves in it, we give a formula relating the linking number of the later to the spectral theory of the Laplace operator acting on differential one forms. As an application, we compute the linking number of any two multi-geodesics of the flat torus of dimension 3, g…
Paper finds shortest geodesic paths on hyperbolic surfaces.
problem Finding the shortest geodesic paths on hyperbolic surfaces.
method Analyzes genus g hyperbolic surfaces to find minimal length geodesics.
result Minimal geodesic length is realized by a specific polygon.
The study of second eigenvalues of hyperbolic surfaces improves bounds and investigates their behavior for large genus.
problem Investigating the second eigenvalues of closed hyperbolic surfaces for large genus.
method Analyzing the shortest length of separating multi-geodesics and investigating the ratio of eigenvalues to this length.
result For large genus, the second eigenvalue of a hyperbolic surface is uniformly comparable to 1/ln(g).