arXiv research

The study examines surfaces in hyperbolic 3-manifolds that become nearly flat.

Estimates geodesics on surfaces without conjugate points.

The paper counts geodesic loops on surfaces without conjugate points.

The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.

The study of random surfaces reveals asymptotic lengths of separating geodesics.

Study of Lévy flights on Zoll surfaces, revealing geometric information.

We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we…

Study counts geodesics on modular surface, linking to necklace counting.

Geodesics of the same type on curved surfaces are randomly distributed.

Study geodesic curvature of logarithmic spirals on curved surfaces.

Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.

Study of large-$n$ asymptotics for Weil-Petersson volumes of hyperbolic surfaces with cusps.

Mirzakhani's thesis counts geodesics on hyperbolic surfaces, finding a specific asymptotic formula.

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.

We find an upper bound for the asymptotic dimension of a hyperbolic metric space with a set of geodesics satisfying a certain boundedness condition studied by Bowditch. The primary example is a collection of tight geodesics on the curve graph of a compact orientable surface. We use this to conclude that a curve graph h…

On a surface with a Finsler metric, we investigate the asymptotic growth of the number of closed geodesics of length less than $L$ which minimize length among all geodesic multicurves in the same homology class. An important class of surfaces which are of interest to us are hyperbolic surfaces.

Study on geodesics on random hyperbolic surfaces, showing variance asymptotic to X log X.

Lengths of simple closed geodesics on hyperbolic surfaces in prescribed homology classes

Study describes frequencies of geodesics on hyperbolic surfaces as genus grows.

Study on geodesics on high genus expander surfaces, proving filling and non-simple properties.

The length of shortest non-simple geodesics grows logarithmically with surface genus.

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length generically has quadratic asymptotics in this length, with a common coefficient c…

The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.

Study asymptotics of Selberg zeta function on spin moduli space.

Defines and analyzes generalized normal ruled surfaces of curves in 3D space.

The study counts geodesics on curved surfaces with specific intersections.

Counting periodic geodesics of bounded length and commutator structure on hyperbolic surfaces.

We present a new construction of embedded minimal surfaces in hyperbolic space with $3$ asymptotically totally geodesic ends and arbitrary finite genus.

Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.

Study geodesics of meromorphic connections on Riemann surfaces.

In this paper, we study the asymptotic geometry of Teichmuller space of Riemann surfaces and give bounds on the Weil-Petersson sectional curvature of Teichmuller space, in terms of the length of the shortest geodesic on the surface. This will also imply that the sectional curvature is not pinched from above or below by…

Characterizes rotational solitons for curve shortening flow on revolution surfaces.

Formula for integrating random variables on hyperbolic surfaces.

Exploiting a relationship between closed geodesics on a generic closed hyperbolic surface S and a certain unipotent flow on the product space T_1(S) x T_1(S), we obtain a local asymptotic equidistribution result for long closed geodesics on S. Applications include asymptotic estimates for the number of pants immersions…

In this paper, we proved the quantum layer over a surface which is ruled outside a compact set, asymptotically flat but not totally geodesic admits ground states.

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.

We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.

Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article …

In this study we give definitions and characterizations of transversal surfaces of timelike ruled surfaces. We study some special cases such as the striction curve is a geodesic, an asymptotic line or a line of curvature. Moreover, we obtain developable conditions for transversal surfaces of a timelike ruled surface.

Counting hyperbolic multi-geodesics with individual component lengths.

Let $d$ be a positive square-free integer $\equiv 3 \pmod{4}$ such that there is no invariant of the ideal class group $\mathbb{Q}\lbrack \sqrt{-d}\rbrack$ which is divisible by $4$. We prove an asymptotic formula for the number of immersed totally geodesic surfaces in $Γ_{-d}\backslash \mathbb{H}^3$ having area less t…

Given a hyperbolic surface and a simple closed geodesic on it, complex-twists along the curve produce a holomorphic family of deformations in Teichmüller space, degenerating to the Riemann surface where it is pinched. We show there is a corresponding Teichmüller disk such that the two are strongly asymptotic, in the Te…

The mapping class group of a surface $§$ acts on the set of closed geodesics on $§$. This action preserves self-intersection number. In this paper, we count the orbits of curves with at most $K$ self-intersections, for each $K \geq 1$. (The case when $K=0$ is already known.) We also restrict our count to those orbits t…

The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.

Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two tangents coincide with a given tangent, then we obtain the normal curvature of t…

Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.

We prove a Gauss-Bonnet formula for the extrinsic curvature of complete surfaces in hyperbolic space under some assumptions on the asymptotic behaviour. The result is given in terms of the measure of geodesics intersecting the surface non-trivially, and of a conformal invariant of the curve at infinity.

We show that the number of square-tiled surfaces of genus $g$, with $n$ marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most $L$ squares, is asymptotic to $L^{6g-6+2n}$ times a product of constants appearing in Mirzakhani's count of …