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.

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.

This paper classifies minimal fillings of lens spaces.

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

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

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.

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 …

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…

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…

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…

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…

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…

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.

The study computes trace fields and minimal polynomials for specific knots and links.

It has been observed that most manifolds in the Callahan-Hildebrand-Weeks census of cusped hyperbolic $3$-manifolds are obtained by surgery on the minimally twisted 5-chain link. A full classification of the exceptional surgeries on the 5-chain link has recently been completed. In this article, we provide a complete cl…

Algorithm finds minimal volume hyperbolic links in 3-manifolds.

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

Paper finds infinite family of minimal triangulations for complex 3D shapes.

Study on Legendrian knots and their non-orientable Lagrangian fillings.

Proves minimization for Kähler manifolds with automorphisms.

Upper bound found for minimal area in Einstein 4-manifolds.

We extend Thurston's metric to projective filling currents, embedding Teichmüller space into the larger space.

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold M^n is bounded from above by (n+2)!d/3,…

In this paper we construct families of homology spheres which bound 4-manifolds with intersection forms isomorphic to $-E_8$. We show that these families have arbitrary large correction terms. This result says that among homology spheres, the difference of the maximal rank of minimal sub-lattice of definite filling and…

The paper proves infinitely many strong symplectic fillings for cusp singularity links.

Computes minimal dilatation for Thurston maps on surfaces.

Study lens spaces' definite fillings, classifying those with specific inequalities.

Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the self-intersecting minimal surface that fills the trefoil knot.

Maximizes filling systems on surfaces with given boundary components.

An irreducible 3--manifold with torus boundary either is a Seifert fibered space or admits at most three lens space fillings according to the Cyclic Surgery Theorem. We examine the sharpness of this theorem by classifying the non-hyperbolic manifolds with more than one lens space filling, classifying the hyperbolic man…

Study the JSJ-decomposition of a specific 3-manifold.

The paper calculates critical points of systole function on Teichmüller space.

Study finds minimum lengths of curves on a one-holed torus.

We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S^2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbi…

The paper studies symplectic operations on Stein fillings of Brieskorn singularities.

A filling Dehn sphere $Σ$ in a closed 3-manifold $M$ is a sphere transversely immersed in $M$ that defines a cell decomposition of $M$. Every closed 3-manifold has a filling Dehn sphere. The Montesinos complexity of a $3$-manifold $M$ is defined as the minimal number of triple points among all the filling Dehn spheres …