arXiv research

Complex surfaces show non-simply connected diffeomorphism groups with non-homotopic loops.

33 curves on a 3-genus surface, all intersecting at most once.

Study shows how many crossings arise in curves on surfaces.

We prove that on a closed surface of genus $g$, the cardinality of a set of simple closed curves in which any two are non-homotopic and intersect at most once is $\lesssim g^2 \log(g)$. This bound matches the largest known constructions to within a logarithmic factor. The proof uses a probabilistic argument in graph th…

In this article we show that for any given Riemann surface $Σ$ of genus $g$, we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to $Σ$ in terms of the genus and the combined extremal lengths on $Σ$ of $(g-1)$ disjoint, non-homotopic, simple closed comp…

Let $S$ be an $n$-punctured sphere, with $n \geq 3$. We prove that $\binom{n}{3}$ is the maximum size of a family of pairwise non-homotopic simple arcs on $S$ joining a fixed pair of distinct punctures of $S$ and pairwise intersecting at most twice. On the way, we show that a square annular diagram $A$ has a corner on …

We prove that on a punctured oriented surface with Euler characteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a cl…

Let $S_{g}$ denote the genus $g$ closed orientable surface. For $k\in \mathbb{N}$, a $k$-system is a collection of pairwise non-homotopic simple closed curves such that no two intersect more than $k$ times. Juvan-Malnič-Mohar \cite{Ju-Mal-Mo} showed that there exists a $k$-system on $S_{g}$ whose size is on the order o…

New surfaces in a 4-manifold are found that are not isotopic but homotopic.

New bounds on curves on torus with few intersections.

New groups formed by twists on surfaces are free.

The study counts 23 maximal 1-systems on a torus with 2 punctures.

Maximizes filling systems on surfaces with given boundary components.

Let $\mathcal {M}$ be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let $cl(Σ^{(1)}_{iness})$ be the closure of the union of all singular knots in $\mathcal {M}$ with exactly one ordinary double point and such that the two resolutions repres…

The purpose of this article is two-fold: We first give a more elementary proof of a recent theorem of Korkmaz, Monden, and the author, which states that the commutator length of the n-th power of a Dehn twist along a boundary parallel curve on a surface with boundary S of genus g at least two is the floor of (|n|+3)/2 …

For non-homotopic maps $u,v\in C^{\infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $γ_p(u,v)$ for which there exist paths $u_t\in W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with $\|du_t\|_{L^p}^p\leq γ_p(u,v)$. When $u$ and $v$ are $(k-2)$-homotopic, work of Hang and Lin shows t…

Let $F_g$ be a closed orientable surface of genus $g$. A set $Ω= \{ γ_1, \dots, γ_s\}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a \emph{filling system} or simply a \emph{filling} of $F_g$, if $F_g\setminus Ω$ is a union of $b$ topological discs for some $b\geq 1$. A filling system is called \em…

Let S be a closed oriented surface of genus at least two. Labourie and the author have independently used the theory of hyperbolic affine spheres to find a natural correspondence between convex RP^2 structures on S and pairs (Σ,U) consisting of a conformal structure Σon S and a holomorphic cubic differential U over Σ. …