arXiv research

The paper constructs minimal coherent filling pairs on surfaces.

This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.

The study finds an upper limit for the number of minimal origami pairs on a surface.

Let $S_g$ be a closed surface of genus $g$ and let $(α, β)$ be a filling pair on $S_g$; then $i(α, β) \geq 2g-1$, where $i$ is the (geometric) intersection number. Aougab and Huang demonstrated that (exponentially many) minimally-intersecting filling pairs exist on $S_g$ when $g > 2$ by a construction w…

In this short note, we construct a minimally intersecting pair of simple closed curves that fill a genus 2 surface with an odd, greater than 3, number of punctures. This finishes the determination of minimally intersecting filling pairs for all surfaces completing the work of Aougab-Huang and Aougab-Taylor.

Let $S_{g}$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill $S_{g}$ and intersect minimally, by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric g…

Computes minimal dilatation for Thurston maps on surfaces.

The article finds conditions for separating filling pairs on surfaces and constructs a Morse function.

Let $F_g$ denote a closed oriented surface of genus $g$. A set of simple closed curves is called a filling of $F_g$ if its complement is a disjoint union of discs. The mapping class group $\text{Mod}(F_g)$ of genus $g$ acts on the set of fillings of $F_g$. The union of the curves in a filling forms a graph on the surfa…

Baker showed that 10 of the 12 classes of Berge knots are obtained by surgery on the minimally twisted 5-chain link. In this article we enumerate all hyperbolic knots in S^3 obtained by surgery on the minimally twisted 5 chain link that realize the maximal known distances between slopes corresponding to exceptional (le…

The paper classifies and studies symplectic and contact properties of circular spherical divisors.

Study on the minimum length of curves on once-punctured hyperbolic surfaces.

We find all $P$-resolutions of quotient surface singularities (especially, tetrahedral, octahedral, and icosahedral singularities) together with their dual graphs, which reproduces Jan Steven's list [Manuscripta Math. 1993] of the numbers of $P$-resolutions of each singularities. We then compute the dimensions and Miln…

The paper proves finiteness of cosmetic fillings on a specific type of 3-manifold.

Measuring foliations at infinity of quasi-Fuchsian manifolds, proving filling pairs and showing realisation.

A pair $(α, β)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $α\cupβ$ in $M_g$ are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. In \cite{Aou}, Aougab-Huang con…

New origamis found for surfaces with minimal intersections.

Origami edge-paths connect coherent curves on surfaces.

We show that if a simple 3-manifold $M$ has two Dehn fillings at distance $Δ\geq 4$, each of which contains an essential annulus, then $M$ is one of three specific 2-component link exteriors in $S^3$. One of these has such a pair of annular fillings with $Δ= 5$, and the other two have pairs with $Δ= 4$.

The study proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.

Study shows connectedness of Bowditch boundary persists in long Dehn fillings.

Let $S$ be a surface of negative Euler characteristic and consider a finite filling collection $Γ$ of closed curves on $S$ in minimal position. An observation of Foulon and Hasselblatt shows that $PT(S) \setminus \hatΓ$ is a finite-volume hyperbolic 3-manifold, where $PT(S)$ is the projectivized tangent bundle and $\ha…

Paper shows regions close to negatively curved metrics are minimal fillings and rigid.

In this paper, we investigate the minimal symplectic fillings of small Seifert 3-manifolds with a canonical contact structure. As a result, we classify all minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, we also demonstrate that every such a minimal symplectic fillin…

Paper finds minimal number of curves in surface systems.

A pair of distinct free homotopy classes of closed curves in an orientable surface $F$ with negative Euler characteristic is said to be length equivalent if for any hyperbolic structure on $F$, the length of the geodesic representative of one class is equal to the length of the geodesic representative of the other clas…

This paper classifies minimal fillings of lens spaces.

The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.

In this article, we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a …

We classify all the non-hyperbolic Dehn fillings of the complement of the chain-link with 3 components, conjectured to be the smallest hyperbolic 3-manifold with 3 cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many 1-cusped and 2-cusped hyperbolic manifolds, including most of tho…

In this paper we investigate the distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing essential small surfaces including non-orientable surfaces. Especially we study the situations where one filling creates an essential sphere or projective plane, and the other creates an essenti…

Simplified presentation of symplectic fillings of lens spaces.

We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blo…

New filling functions for groups with coefficients show different asymptotic behavior.

The paper characterizes symplectic fillings of Seifert 3-manifolds using rational blowdowns.

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…

We prove that every minimal symplectic filling of the link of a quotient surface singularity can be obtained from its minimal resolution by applying a sequence of rational blow-downs and symplectic antiflips. We present an explicit algorithm inspired by the minimal model program for complex 3-dimensional algebraic vari…

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show…

This paper states a formula for the difference of the Holmes-Thompson volumes of two simple Finsler manifolds of arbitrary dimension, in terms of the boundary distances and their derivatives. An application is a preconditioned filling minimality result.

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this r…

We consider in this paper the minimally twisted chain link with 5 components in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn fillings on its exterior M5. The 3-manifold M5 is a nicely symmetric hyperbolic one, filling which one gets a wealth of hyperbolic 3-manifolds having 4 or fewer (includi…

The paper studies properties of optimal metrics associated to curves on surfaces.

In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let $M$ be such a knot manifold and let $β$ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $M$ with slope $α$ produces a Seifert fibred manifold,…

This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above mentioned paper.

The paper fills hyperbolic surfaces with a minimal number of systoles.

Minimal triangulations for 229 hyperbolic census knots discovered.

Study connects lens spaces' fundamental group to their symplectic fillings' second Betti numbers.

Paper explores using 0-surgery to find exotic 4-manifolds.