arXiv research

Max systoles on spheres with punctures are counted.

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

Unique maximal curve systems found for up to 5 punctures.

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

In this paper, we will prove a result of nonexistence on harmonic diffeomorphisms between punctured spaces. In particular, we will given an elementary proof to the nonexistence of rotationally symmetric harmonic diffeomorphisms from the punctured Euclidean space onto the punctured hyperbolic space.

In this paper, we calculate the p-torsion of the Farrell cohomology for low genus pure mapping class groups with punctures, where p is an odd prime. Here, `low genus' means g=1,2,3; and `pure mapping class groups with punctures' means the mapping class groups with any number of punctures, where the punctures are not al…

Researchers compute TQFT representation for sphere with 4 punctures.

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

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

Describes curves on surfaces with punctures and boundaries.

We generalize Dynnikov coordinate system previosly defined on the standard punctured disk to an orientable surface of genus-1 with n punctures and one boundary component.

Let $h$ be a complete metric of Gaussian curvature $K_0$ on a punctured Riemann surface of genus $g \geq 1$ (or the sphere with at least three punctures). Given a smooth negative function $K$ with $K=K_0$ in neighbourhoods of the punctures we prove that there exists a metric conformal to $h$ which attains this function…

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.

The study counts curves on a once-punctured torus with self-intersections.

Study shows Bergman kernel quotient approaches one for punctured surfaces.

Sharp bounds found on shortest geodesic on punctured spheres.

Researchers determine all possible representations of monodromy for Schwarzian equations on punctured surfaces.

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

We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to de…

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

Study of Fubini-Study forms on surfaces with punctures.

Based on the Kauffman bracket at $A=e^{i π/4}$, we defined an invariant for a special type of $n$-punctured ball tangles. The invariant $F^n$ takes values in the set $PM_{2\times2^n}(\mathbb Z)$ of $2\times 2^n$ matrices over $\mathbb Z$ modulo the scalar multiplication of $\pm1$. We provide the formula to compute the …

In this paper we give a necessary and sufficient condition in which a sequence of Kleinian punctured torus groups converges. This result tells us that every exotically convergent sequence of Kleinian punctured torus groups is obtained by the method due to Anderson and Canary. Thus we obtain a complete description of th…

Presented an algebra structure for a specific geometric surface.

The paper resolves kinks on curves on surfaces with punctures.

Paper presents skein algebras for spheres with punctures.

Using the techniques developed in \cite{SunSun}, we give estimations of the Bergman kernel of the punctured disk with the standard complete Poincaré metric. As an application, we improve the result of \cite{AMM} on the Bergman kernels of punctured Riemann surfaces near singularities.

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

Study unbounded $\mathfrak{sl}_3$-laminations around punctures.

Study finds bounds for systole length on arithmetic punctured spheres.

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.

Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free t…

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

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincaré metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the p…

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

New theorem on spheres with punctures using infinity metric.

New proof for stable reduction theorem using Kähler-Einstein metrics.

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…

We determine the non-null homologous knots in lens spaces whose exteriors contain properly embedded once-punctured tori. All such knots arise as surgeries on the Whitehead link and are grid number 1 in their lens spaces. As a corollary, we classify once-punctured torus bundles that admit a lens space filling.

Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.

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

Extends harmonic maps compactification to punctured Riemann surfaces.

Formula calculates index for CR operators on surfaces with boundary punctures.

Studies acceptable bundles on a partially punctured polydisk.

Extending the Labourie-Loftin correspondence, we establish, on any punctured oriented surface of finite type, a one-to-one correspondence between convex projective structures with specific types of ends and punctured Riemann surface structures endowed with meromorphic cubic differentials whose poles are at the puncture…

New metrics with constant Q-curvature created by gluing.

Study bounds topological entropy of maps on surfaces with punctures based on mapping torus homology.