arXiv research

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

This work classifies belted sum decompositions of fully augmented links.

This paper constructs a family of constant mean curvature immersions of the thrice-punctured Riemann sphere into Euclidean 3-space with asymptotically Delaunay ends via loop group methods.

The strip map is a natural map from the arc complex of a bordered hyperbolic surface $S$ to the vector space of infinitesimal deformations of $S$. We prove that the image of the strip map is a convex hypersurface when $S$ is a surface of small complexity: the punctured torus or thrice punctured sphere.

We show that an immersed thrice-punctured sphere in a cusped orientable hyperbolic 3-manifold is either embedded or has a single clasp in a manifold obtained by hyperbolic Dehn filling on a cusp of the Whitehead link complement.

Study uses Dynnikov coordinates to analyze actions of Dehn twists on a thrice-punctured disc.

This paper determines which orientable hyperbolic 3-manifolds contain simple closed geodesics. The Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that th…

Constructs hyperbolic affine spheres and Calabi-Yau metrics.

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Go…

The action of the mapping class group of the thrice-punctured projective plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corres…

Lower bounds on geodesic length with few intersections on hyperbolic surfaces.

We present a theorem on the unitarizability of loop group valued monodromy representations and apply this to show the existence of new families of constant mean curvature surfaces homeomorphic to a thrice-punctured sphere in the simply-connected 3-dimensional space forms $\R^3$, $\bbS^3 $ and $\bbH^3$. Additionally, we…

New surfaces found in 5D space.

The study connects polygon areas and projective structures in 3D space.

An explicit formula for the generalized hyperbolic metric on the thrice--punctured sphere $¶\backslash \{z_1, z_2, z_3\}$ with singularities of order $α_j \le 1$ at $z_j$ is obtained in all possible cases $α_1+α_2+α_3 >2$. The existence and uniqueness of such a metric was proved long time ago by Picard \cite{Pic1905} a…

Mazur's knot exterior is described by a single regular ideal octahedron, leading to hyperbolic structures related to the Whitehead link.

Let L --> X be a complex line bundle over a compact connected Riemann surface. We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle has parabolic structure. These conditions appear naturally in the study of vortex …

Denote the free group on two letters by F2 and the SL(3,C)-representation variety of F2 by R = Hom(F2, SL(3, C)). There is a SL(3,C)-action on the coordinate ring of R, and the geometric points of the subring of invariants is an affine variety X. We determine explicit minimal generators and defining relations for the s…

We prove that there is a true asymptotic formula for the number of one sided simple closed curves of length $\leq L$ on any Fuchsian real projective plane with three points removed. The exponent of growth is independent of the hyperbolic structure, and it is noninteger, in contrast to counting results of Mirzakhani for…

The study finds infinitely many twist knot complements with totally geodesic surfaces.

New examples of surface bundles found over surfaces.

In this paper we investigate the higher dimensional divergence functions of mapping class groups of surfaces and of CAT(0)--groups. We show that, for mapping class groups of surfaces, these functions exhibit phase transitions at the rank (as measured by thrice the genus plus the number of punctures minus 3). We also pr…

Functorial approach connects operads to Lie bialgebras.

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.