arXiv research

The study finds at least two short, simple geodesic chords on a disk with convex boundary.

We provide a combinatorial condition characterizing curves that are short along a Teichmueller geodesic. This condition is closely related to the condition provided by Minsky for curves in a hyperbolic 3-manifold to be short. We show that short curves in a hyperbolic manifold homeomorphic to S x R are also short in the…

The paper bounds eigenvalue multiplicities for hyperbolic surfaces using short geodesics.

The paper calculates the volume growth of hyperbolic surfaces with short geodesics.

The Weil-Petersson metric's curvature vanishes for surfaces with short geodesics.

A closed Teichmuller geodesic in the moduli space M_g of Riemann surfaces of genus g is called L-short if it has length at most L/g. We show that, for any L > 0, there exist e_2 > e_1 > 0, independent of g, so that the L-short geodesics in M_g all lie in the intersection of the e_1-thick part and the e_2-thin part. We …

Study the energy distribution of harmonic 1-forms on Riemann surfaces with a short geodesic.

Study bounds changes in hyperbolic 3-manifold structures after drilling short geodesics.

For odd-dimensional spheres, there's always a second short geodesic.

Method constructs WP geodesics to study Teichmüller space behavior.

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…

Geodesically convex functions are continuous on Riemannian manifolds.

Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first $3g-3$ geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short ($< \log (g)$) geodesics on a …

Given a geometrically finite hyperbolic cone-manifold, with the cone singularity sufficiently short, we construct a one parameter family of cone-manifolds decreasing the cone angle to zero. We also control the geometry of this one parameter family via the Schwarzian derivative of the projective boundary and the length …

3D spheres can't be swept by short curves, complicating geodesic length estimates.

Formula removes geometric patterns from random hyperbolic surfaces.

A short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesics

Study on geodesics and eigenvalues on random hyperbolic surfaces with cusps.

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…

Study geodesics on random hyperbolic surfaces, finding variance similar to prime number theory.

In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration.

This expository article discusses some connections between the geometry of a hyperbolic 3-manifold homotopy-equivalent to a surface, and the combinatorial properties of its end invariants. In particular a necessary and sufficient condition is stated for the manifold to have arbitrarily short geodesics, in terms of a se…

Lower bounds on geodesic lengths for spheres with Willmore energy.

Totally geodesic sections found in polar actions.

This short survey illustrates the ideas of Teichmuller dynamics. As a model application we consider the asymptotic topology of generic geodesics on a "flat" surface and count closed geodesics and saddle connections. This survey is based on the joint papers with A.Eskin and H.Masur and with M.Kontsevich.

Study proves convergence of quantized geodesics to Mabuchi geodesics.

Proofs contractibility of geodesic triangulations spaces and non-trivial homotopy groups.

Researchers found spiraling conformal geodesics in 3D space.

3D manifold with positive curvature has a short geodesic.

New method uses short geodesics to approximate marked length spectrum.

In this paper we construct quasiconformal embeddings from Y-pieces that contain a short boundary geodesic into degenerate ones. These results are used in a companion paper to study the Jacobian tori of Riemann surfaces that contain small simple closed geodesics.

Short note on upper bounds for loop homology classes.

For two measured laminations $ν^+$ and $ν^-$ that fill up a hyperbolizable surface $S$ and for $t \in (-\infty, \infty)$, let $L_t$ be the unique hyperbolic surface that minimizes the length function $e^t l(ν^+) + e^{-t} l(ν^-)$ on Teichmuller space. We characterize the curves that are short in $L_t$ and estimate their…

Study on shortest geodesics crossing multiple times on hyperbolic surfaces with cusps.

We give a combinatorial proof, using the hyperbolicity of the curve graphs, of the bounded geodesic image theorem of Masur and Minsky. Recently it has been shown that curve graphs are uniformly hyperbolic, thus a universal bound can be given for the diameter of the geodesic image. We also generalize the theorem for pro…

We prove the absence of a universal diameter bound on lengths of curves in a sweep-out of a Riemannian 2-sphere. If such bound existed it would yield a simple proof of existence of short geodesic segments and closed geodesics on a sphere of small diameter.

If $M$ is a finite volume complete hyperbolic $3$-manifold, the quantity $\mathcal A_1(M)$ is defined as the infimum of the areas of closed minimal surfaces in $M$. In this paper we study the continuity property of the functional $\mathcal A_1$ with respect to the geometric convergence of hyperbolic manifolds. We prove…

New bounds on shortest geodesic loops on a sphere.

A subset of the sphere is said short if it is contained in an open hemisphere. A short closed set which is geodesically convex is called a cap. The following theorem holds: 1. The minimal number of short closed sets covering the $n$-sphere is $n+2$. 2. If $n+2$ short closed sets cover the $n$-sphere then (i) their inte…

New families of non-singular geodesic orbit nilmanifolds discovered.

For each $g \ge 2$, we prove existence of a computable constant $ε(g) > 0$ such that if $S$ is a strongly irreducible Heegaard surface of genus $g$ in a complete hyperbolic 3-manifold $M$ and $γ$ is a simple geodesic of length less than $ε(g)$ in $M$, then $γ$ is isotopic into $S$.

Geodesic X-ray transform proves injective for smooth one-forms on gas giant manifolds.

Study on ordering geodesics on hyperbolic surfaces.

The study bounds the number of closed geodesics in a specific orbit closure of surfaces.

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp sha…

Entropy measures geodesic flow complexity.

Derives generalizations of the long neck principle and spectral width inequality.

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We pr…