arXiv research

Describes curves on surfaces with punctures and boundaries.

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

Study cohomology of surfaces with punctures and boundaries, proving bounds on rational cohomology.

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.

The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.

We report on the computation of the integral homology of the mapping class group of genus g surfaces with one boundary curve and m punctures, when 2g + m is smaller than 6. In particular, it includes the genus 2 case with no or one puncture.

In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent--Leininger--Schleimer and Mitra, we construct a universal Cannon--Thurston map from a subset of the circle at infinity for the closed surfac…

The paper provides presentations for mapping class groups and cluster automorphism groups of surfaces.

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. Three interesting fe…

We classify incompressible, boundary-incompressible, nonorientable surfaces in punctured-torus bundles over $S^1$. We use the ideas of Floyd, Hatcher, and Thurston. The main tool is to put our surface in the "Morse position" with respect to the projection of the bundle into the basis S^1.

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has non-compact ends with boundary cusps. This extends Wolf's pa…

It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-d…

We obtain simple generating sets for various mapping class groups of a nonorientable surface with punctures and/or boundary. We also compute the abelianizations of these mapping class groups.

In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics…

We compare some natural triangulations of the Teichmüller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt Scannell-Wolf's proof to show that grafting semi-infinite cylinders at the ends of hyperbolic surfaces with fixed boundary lengths is a homeomorphism. This way, we construct…

Study homology groups for non-orientable surfaces with boundary.

Consider an oriented compact surface F of positive genus, possibly with boundary, and a finite set P of punctures in the interior of F, and define the punctured mapping class group of F relatively to P to be the group of isotopy classes of orientation-preserving homeomorphisms h: F-->F which pointwise fix the boundary …

We show that the center of the Goldman algebra associated to a closed oriented hyperbolic surface is trivial. For a hyperbolic surface of finite type with nonempty boundary, the center consists of closed curves which are homotopic to boundary components or punctures.

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…

Boundary Dehn twists become trivial after abelianization.

We study two $2$-dimensional Teichmüller spaces of surfaces with boundary and marked points, namely, the pentagon and the punctured triangle. We show that their geometry is quite different from Teichmüller spaces of closed surfaces. Indeed, both spaces are exhausted by regular convex geodesic polygons with a fixed numb…

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

Let $M$ be the moduli space of rank 3 parabolic vector bundles over a Riemann surface with several punctures. By the Mehta-Seshadri correspondence, this is the space of rank 3 unitary representations of the fundamental group of the punctured surface with specified conjugacy classes of the images of each boundary compon…

Study on dimensions of mapping class groups of non-orientable surfaces.

We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and hyperbolic two-bridge knots.

Boundary Dehn twist on $K3$ surfaces becomes trivial after abelianization.

We study the cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of $\operatorname{Ext}^1$ as intersection numbers of ta…

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…

We prove that the Teichmüller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichmüller metric) is almost isometric to the Teichmüller space of punctured surfaces equipped with the Thurston metric (resp. the Teichmüller metric).

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher. We conver…

The paper studies how hyperbolic surfaces degenerate along harmonic map rays.

We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear repres…

We show that the Teichmüller space of a surface without boundary and with punctures, equipped with Thurston's metric is the limit (in an appropriate sense) of Teichmüller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the tran…

In this paper we prove that a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface $\overline{M}$ with boundary punctured in a finite number of interior points and that $M$ can be represented in terms of meromorphic …

The Dynnikov coordinate system puts global coordinates on the boundary of Teichmüller space of an $n$--punctured disk. We survey the Dynnikov coordinate system, and investigate how we use this coordinate system to study pseudo--Anosov braids making use of results from Thurston's theory on surface homeomorphisms.

We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative …

This paper studies deformations of hyperbolic surfaces with special structures.

The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2π. Through every point in such a metric there is a geodesic that saturates the length condition, and saturating geodesics …

Homological mirror symmetry proved for symmetric squares of punctured spheres.

In this paper we complete the topological description of the space of representations of the fundamental group of a punctured surface in SL(2,R) with prescribed behavior at the punctures and nonzero Euler number, following the strategy employed by Hitchin in the unpunctured case and exploiting Hitchin-Simpson correspon…

For a closed 4-manifold $X$ and a knot $K$ in the boundary of punctured $X$, we define $γ_X^0(K)$ to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured $X$ with boundary $K$. Note that $γ^0_{S^4}$ is equal to the non-orientable 4-ball genus and hence $γ^0_X$ is a generalizati…

In a recent paper A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces $Σ\_n$ in $B^3$ which have genus $0$ and $n$ boundary components, for all $n \geq 3$. For large $n$, we give an independent construction of $Σ\_n$ and prove the existence of free boundary minimal surfaces $\tilde Σ\_n…

We describe triangle coordinates for integral laminations on a non-orientable surface $N_{k,n}$ of genus $k$ with $n$ punctures and one boundary component, and give an explicit bijection from the set of integral laminations on $N_{k,n}$ to $(\mathbb{Z}^{2(n+k-2)}\times \mathbb{Z}^k)\setminus \left\{0\right\}$.

We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface $\mathbf{S}$ with marked points and non-empty boundary, which generalizes Brüstle-Zhang's result for the puncture free case. As an application, we show that the intersection of the shi…

This paper extends the decorated Teichmüller theory developed before for punctured surfaces to the setting of ``bordered'' surfaces, i.e., surfaces with boundary, and there is non-trivial new structure discovered. The main new result identifies the arc complex of a bordered surface up to proper homotopy equivalence wit…

The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.

A smooth end of a Bryant surface is a conformally immersed punctured disc of mean curvature 1 in hyperbolic space that extends smoothly through the ideal boundary. The Bryant representation of a smooth end is well defined on the punctured disc and has a pole at the puncture. The Willmore energy of compact Bryant surfac…

Let T(N) be the subgroup of the mapping class group of a nonorientable surface N (possibly with punctures and/or boundary components) generated by twists about two-sided circles. We obtain a simple generating set for T(N). As an application we compute the first homology group (abelianization) of T(N).