arXiv research

Study non-fillable curves in a hyperbolic surface with a real line.

The paper studies a flow of Legendre curves, generalizing the inverse curvature flow of regular curves.

We analyzed the problem of finding a surfaces family through an asymptotic curve with Cartan frame. We obtain the parametric representation for surfaces family whose members have the same as an asymptotic curve. By using the Cartan frame of the given null curve, we present the surface as a linear combination of this fr…

Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.

In the present paper, we handle the problem of finding a hypersurface family from a given asymptotic curve in R^4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. …

Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve has attracted much interest. In the present paper, we propose a new method to construct a surface interpolating a given curve as the asymptotic curve of it. Also, we analyze the co…

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

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…

Solves area-minimizing surface problem for finite curves in H^2xR.

In this paper we prove a universal inequality describing the asymptotic behavior of support points for planar continuous curves. As corollaries we get an analogous result for tangent points of differentiable planar curves and some (partially known) assertions on the asymptotic of the mean value points for various class…

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\mathbb{R}^2$ under the $α$-curve shortening flow for exponents $α>\frac12$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $α$-curve shortening flow to the …

Characterizes rotational solitons for curve shortening flow on revolution surfaces.

We provide a detailed description of solutions of Curve Shortening in $\R^n$ that are invariant under some one-parameter symmetry group of the equation, paying particular attention to geometric properties of the curves, and the asymptotic properties of their ends. We find generalized helices, and a connection with curv…

In this paper, we express surfaces parametrically through a given spacelike (timelike) asymptotic curve using the Frenet frame of the curve in Minkowski 3-space. Necessary and sufficient conditions for the coefficients of the Frenet frame to satisfy both parametric and asymptotic requirements are derived. We also prese…

The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.

In this paper, we show that the minimal asymptotic translation length of the Torelli group $\mathcal{I}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group $Mod(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translat…

Solves asymptotic $A_n$-realization problem for curves.

Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.

Study on closed curves on negatively curved surfaces, linking number formula, and restrictions.

Metrics are semipositively curved if they meet a specific asymptotic condition.

We prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves, and of Hasset on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solved a questi…

$\mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. In this paper we study the asymptotics of $\mathcal{H}-$holomorphic curves defined on a sequence of degenerating cylinders.

Study on helix curves and their Möbius energy asymptotics.

We prove that a class of asymptotically nonnegatively curved manifolds (in the sense of Abresch) satisfying some uniform Euclidean type volume growth conditions contains only finitely many homeomorphism types.

We give a bound, linear in the complexity of the surface, on the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.

The grand arc graph's asymptotic dimension is shown to be infinite.

Study minimal translation lengths on curve complexes, providing bounds and constructing examples.

Study of height jumps in Ceresa cycle using asymptotic Hodge theory.

Study curvature and torsion from cross-ratios in discrete curves.

In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.

In this paper, we analyze the asymptotic behavior of $κ$-noncollapsed and positively curved steady Ricci solitons and prove that any $n$-dimensional $κ$-noncollapsed steady Kähler-Ricci soliton with non-negative sectional curvature must be flat.

The paper examines the geometry of a curve's centre symmetry set.

Curves become nearly circular over time without initial assumptions.

We study the asymptotic behavior of the asymptotic translation lengths on the curve complexes of pseudo-Anosov monodromies in a fibered cone of a fibered hyperbolic 3-manifold $M$ with $b_1(M) \geq 2$. For a sequence $(Σ_n, ψ_n)$ of fibers and monodromies in the fibered cone, we show that the asymptotic translation len…

Compact, non-convex curve flows are created.

Curved metrics on Wallach spaces bounded by curves under flow.

The paper examines asymptotic lines of plane fields in 3D space.

We show that asymptotically the first Betti number, or the arithmetic genus, of a Shimura curve satisfies the Gauss--Bonnet equality. We also show that the first Betti number of a congruence hyperbolic 3--orbifold asymptotically vanishes relatively to hyperbolic volume.

This paper corrects an error in [Keller-Ressel, M. and Steiner T. "Yield curve shapes and the asymptotic short rate distribution in affine one-factor models." Finance and Stochastics 12.2 (2008): 149-172]. The error concerns the correct expression for the boundary between normal and humped yield curve behavior in affin…

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

The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes span…

The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.

We analyze learning curves of RF models with convex regularization and derive precise asymptotic expressions.

The curve graph and related graphs are hyperbolic and have quasi-tree fibers.

We determine the asymptotic behavior of the optimal Lipschitz constant for the systole map from Teichmuller space to the curve complex.

The study examines complex tangles in Curve Shortening Flow singularities.

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