arXiv research

The paper calculates super Weil-Petersson volumes for large genus.

Study on frequencies of non-simple curves in surfaces of large genus.

Novel approach for large genus intersection number asymptotics.

Study of random multicurves and square-tiled surfaces on large genus surfaces.

The paper calculates large genus limits for quadratic differential volumes and constants.

We extend asymptotic formulas for saddle connections on translation surfaces.

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

We state conjectures on the asymptotic behavior of the Masur-Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel-Veech constants as the genus tends to infinity.

Existence proof of noncompact self-shrinkers with arbitrary genus.

We study the asymptotic behavior of Masur-Veech volumes as the genus goes to infinity. We show the existence of a complete asymptotic expansion of these volumes that depends only on the genus and the number of singularities. The computation of the first term of this asymptotics expansion was a long standing problem. Th…

The study improves the upper bound for the first eigenvalue of Laplacian on compact surfaces of large genus.

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

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

In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volume $ν_1 \big( \mathcal{H}_1 (m) \big)$ of a stratum i…

New expanders for mean curvature flow contradict genus-reduction conjecture.

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

In this paper we consider the large genus asymptotics for two classes of Siegel-Veech constants associated with an arbitrary connected stratum $\mathcal{H} (α)$ of Abelian differentials. The first is the saddle connection Siegel-Veech constant $c_{\text{sc}}^{m_i, m_j} \big( \mathcal{H} (α) \big)$ counting saddle conne…

In this article we study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in any thick part of moduli space of Riemann surfaces of genus $g$ is uniformly comp…

Let $S_g$ be a closed surface of genus $g$ and $\mathbb{M}_g$ be the moduli space of $S_g$ endowed with the Weil-Petersson metric. In this paper we investigate the Weil-Petersson curvatures of $\mathbb{M}_g$ for large genus $g$. First, we study the asymptotic behavior of the extremal Weil-Petersson holomorphic sectiona…

Study asymptotics of one part monotone Hurwitz numbers in high genus.

We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermo…

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi-classes supported on the boundary cycles of the Deligne-Mumford compactification of the moduli space of curves. Our formulae …

Constructs self-expanders of positive genus for cones in R^3.

Let M be a closed orientable 3-manifold with a negatively curved Riemannian metric. Let {M_i} be a collection of finite regular covers with degree d_i. (1) If the Heegaard genus of M_i grows more slowly than the square root of d_i, then M_i has positive first Betti number for all sufficiently large i. (2) The strong He…

We prove that square-tiled surfaces having fixed combinatorics of horizontal cylinder decomposition and tiled with smaller and smaller squares become asymptotically equidistributed in any ambient linear $GL(\mathbb R)$-invariant suborbifold defined over $\mathbb Q$ in the moduli space of Abelian differentials. Moreover…

We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Che…

Exact asymptotic value of Weil-Petersson volumes computed for large genus surfaces.

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

The ends of a complete embedded minimal surface of {\em finite total curvature} are well understood (every such end is asymptotic to a catenoid or to a plane). We give a similar characterization for a large class of ends of {\em infinite total curvature}, showing that each such end is asymptotic to a helicoid. The resu…

Counting meanders on surfaces of arbitrary genus, with precise asymptotics.

There exists a properly embedded minimal surface of genus one with one end. The end is asymptotic to the end of the helicoid. This genus one helicoid is constructed as the limit of a continuous one-parameter family of screw-motion invariant minimal surfaces--also asymptotic to the helicoid--that have genus equal to one…

New model calculates logarithmic surface diameter.

We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.

New model shows average genus of 2-bridge knots grows linearly with crossing number.

Constructs self-shrinkers with unique asymptotic behavior.

Study on hyperbolic surfaces' volumes, proving asymptotic expansion for high genus.

The study calculates the average genus of 2-bridge knots based on their crossing numbers.

We show that the asymptotic dimension of a hyperbolic relatively hyperbolic graph is finite provided that this holds true uniformly for the peripheral subgraphs and for the electrifiation. We use this to show that the asymptotic dimension of the disk graph of a handlebody of genus at least two is at most quadratic in t…

Paper studies asymptotic dimension and Assouad-Nagata dimension of graphs and surfaces.

Randomly glued tetrahedra form connected 3-manifolds with a single boundary.

We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact with one end. Each has $4g+4$ symmetries and comes from desingularizing the inters…

Study on the mass of Brownian loops on Riemann surfaces as genus grows.

Cayley cones in the octonions $\mathbb{O}$ that are ruled by oriented 2-planes are equivalent to pseudoholomorphic curves in the Grassmannian of oriented 2-planes G(2,8). The well known twistor fibration $G(2,8) -> S^6$ is used to prove the existence of immersed higher-genus pseudoholomorphic curves in $\gro$. Equivale…

Random representations of surface groups approach asymptotic freeness in large $n$ limit.

We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an ana…

For each discriminant $D>1$, McMullen constructed the Prym-Teichmüller curves $W_D(4)$ and $W_D(6)$ in $\mathcal{M}_{3}$ and $\mathcal{M}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of or…

Study minimizes Willmore energy with constraints on surface properties.

Minimal surfaces in a ball have limited area.