arXiv research

Max systoles on spheres with punctures are counted.

Classifies arcs on a 4-punctured sphere that intersect at most once.

Researchers compute TQFT representation for sphere with 4 punctures.

Sharp bounds found on shortest geodesic on punctured spheres.

Researchers compute quantum invariant for four-puncture sphere, verifying volume conjecture.

Study Agol cycles on 2-punctured torus and 5-punctured sphere, finding new dilatation formula.

We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.

Classifies finite orbits of mapping class group action on character varieties.

Presented an algebra structure for a specific geometric surface.

New theorem on spheres with punctures using infinity metric.

Study classifies metrics on a twice-punctured sphere, proving Delaunay metrics are complete.

Study finds bounds for systole length on arithmetic punctured spheres.

Study contact structures on four-punctured spheres, finding infinitely many overtwisted monodromies.

We study the existence or not of harmonic diffeomorphisms between certain domains in the Euclidean 2-sphere. In particular, we show harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for m…

In this paper, we study punctured spheres in two dimensional ball quotient compactifications $(X, D)$. For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded $3$-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness o…

Study on representations of four-punctured sphere group in hyperbolic spaces.

Paper presents skein algebras for spheres with punctures.

Study on rank 2 Higgs bundles on 5-punctured sphere, proving $P=W$ conjecture in lowest degree.

We prove a strong form of finite rigidity for pants graphs of spheres. Specifically, for any $n\geq4$, we construct a finite subgraph $X_n$ of the pants graph $P(S_{0,n})$ of the n-punctured sphere $S_{0,n}$ with the following property. Any simplicial embedding of $X_n$ into any pants graph $P(S_{0,m})$ of a punctured …

We prove that the ending lamination space of the five-punctured sphere is homeomorphic to the Noebeling curve.

We give a new proof that the completion of the Weil-Petersson metric on Teichmüller space is Gromov-hyperbolic if the surface is a five-times punctured sphere or a twice-punctured torus. Our methods make use of the synthetic geometry of the Weil-Petersson metric.

In this paper, we characterize non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. We also show the existence of embeddings of some multibranched surfaces in the 3-sphere which satisfy some homological conditions to be embedded in the…

Researchers prove positivity of skein algebra structure constants for specific surfaces.

Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.

The study constructs new minimal surfaces with more ramified values than previously known.

Homological mirror symmetry proved for symmetric squares of punctured spheres.

We classify the topological types for the unions of the totally geodesic 3-punctured spheres in orientable hyperbolic 3-manifolds. General types of the unions appear in various hyperbolic 3-manifolds. Each of the special types of the unions appears only in a single hyperbolic 3-manifold or Dehn fillings of a single hyp…

Compact character varieties of punctured spheres are proven.

Study finds a minimum volume for vector fields on a punctured sphere.

In these short notes we characterize the loxodromic unit vector fields on antipodally punctured Euclidean spheres as the only ones achieving a lower bound for the volume functional depending on the Poincaré indexes around their singularities.

We derive a precise asymptotic expansion of the complete Kähler-Einstein metric on the punctured Riemann sphere with three or more omitting points. By using Schwarzian derivative, we prove that the coefficients of the expansion are polynomials on the two parameters which are uniquely determined by the omitting points. …

We provide a presentation of the Roger and Yang's Kauffman bracket arc algebra for the once-punctured torus and punctured spheres with three or fewer punctures.

Minimal surfaces in S3(2) linked to vector fields on punctured sphere.

We give counterexamples to a question of Bowditch that if a non-elementary type-preserving representation $ρ:π_1(Σ_{g,n})\rightarrow PSL(2;\mathbb R)$ of a punctured surface group sends every non-peripheral simple closed curve to a hyperbolic element, then must $ρ$ be Fuchsian. The counterexamples come from relative Eu…

Let p be a puncture of a punctured sphere, and let Q be the set of all other punctures. We prove that the maximal cardinality of a set of arcs pairwise intersecting at most once, which start at p and end in Q, is |X|(|X| + 1). We deduce that the maximal cardinality of a set of arcs with arbitrary endpoints pairwise int…

Researchers find a method to construct projective structures on a specific surface.

New metrics with constant Q-curvature created by gluing.

Random hyperbolic surfaces with punctures converge to the Brownian sphere.

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured sphere together with a map to the integers.…

This is the first of at least two articles that describe the moduli spaces of pseudoholomorphic, multiply punctured spheres in R x (S^1 x S^2) as defined by a certain natural pair of almost complex structure and symplectic form. This article proves that all moduli space components are smooth manifolds. Necessary and su…

Hyperbolic links in handlebodies can be composed, unlike in 3-sphere.

We calculate the virtually-cyclic dimension of the mapping class group of a sphere with at most six punctures. As an immediate consequence, we obtain the virtually-cyclic dimension of the mapping class group of the twice-holed torus and of the closed genus-two surface.

Let $Γ$ be a 3-dimensional Kleinian punctured torus group with ccidental parabolic transformations. The deformation space of $Γ$ in the group of Möbius transformations on the 2-sphere is well-known as the Maskit slice of punctured torus groups. In this paper, we study deformations $Γ'$ of $Γ$ in the group of Möbius tra…

Research extends geodesic length function study to three holed sphere.

This paper gives a new obstruction for ribbon-move equivalence of 2-knots. Let $K$ and $K'$ be 2-knots. Let $K$ and $K'$ are ribbon-move equivalent. One corollary to our main theorem is as follows. A 2-dimensional fibered knot whose fiber is the punctured 3-dimensional torus is not ribbon-move equivalent to any 2-dimen…

We show that essential punctured spheres in the complement of links with distance three bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bri…

New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.

The paper proves compactness of metrics with isolated singularities on a sphere.