arXiv research

Study smooth manifolds using disc-presheaves.

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.

Modified proof constructs dual spheres for 4-manifolds.

Study horizontal discs in fat distributions, proving their existence.

A manifold is T-embedded into an affine space if its tangent spaces at distinct points are disjoint. We prove that an n-dimensional disc cannot be T-embedded into 2n-dimensional space.

For a given embedded Lagrangian in the complement of a complex hypersurface we show existence of a holomorphic disc in the complement having boundary on that Lagrangian.

Study detects if a circuit bounds a disc using curve intersections.

We construct knotted proper holomorphic embeddings of the unit disc in C^2.

We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we prove disc embedding results for height 2 and 3 Casson towers which are embedded into a 4-manifold, with some additional fundamental group assumptions. In the proofs we create a capped grope from a Casson towe…

Solves Skopenkov's problem on graph embedding criteria.

The smoothing theory is revised to generalize to different disc embedding spaces.

We show that the open unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ $(n>1)$ admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs.

New minimal discs and annuli found in ellipsoids.

In the Euclidean unit three-ball, we construct compact, embedded, two-sided free boundary minimal surfaces with connected boundary and prescribed high genus, by a gluing construction tripling the equatorial disc. Aside from the equatorial disc itself, these are the first examples in the three-ball of compact free bound…

For a certain maximal unipotent family of Abelian varieties over the punctured disc, we show that after a base change, one can complete the family over a disc such that the whole degeneration can be simultaneously balanced embedded into a projective space by the theta functions. Then we study the relationship between t…

An embedding of the m-times punctured disc into the n-times punctured disc, for n>m, yields an embedding of the braid group on m strands B_m into the braid group on n strands B_n, called a geometric embedding. The main example consists of adding n-m trivial strands to the right of each braid on m strands. We show that …

We give an alternative proof of a theorem of Honda-Kazez-Matić that every non-right-veering open book supports an overtwisted contact structure. We also study two types of examples that show how overtwisted discs are embedded relative to right-veering open books.

Defines a distance function on a manifold using symplectic embeddings and recovers the metric.

In this paper we construct a properly embedded holomorphic disc in the unit ball $\mathbb{B}^2$ of $\mathbb{C}^2$ having a surprising combination of properties: on the one hand, it has finite area and hence is the zero set of a bounded holomorphic function on $\mathbb{B}^2$; on the other hand, its boundary curve is eve…

The topological underpinnings are presented for a new algorithm which answers the question: `Is a given knot the unknot?' The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and …

In this article, we study a free boundary isometric embedding problem for abstract Riemannian two-manifolds with the topology of the disc. Under the assumption of positive Gauss curvature and geodesic curvature of the boundary being equal to one, we show that any such disc may be isometrically embedded into the Euclide…

New phenomena in 4-manifolds show discs with special properties.

Exposes two methods for constructing flat surfaces in 4D spaces.

New complex structure on hyperbolic disc within hyperkaehler space.

Detecting exotic spheres involves analyzing framed configuration spaces.

New approach to extremal hyperbolic surfaces using NEC groups.

Study of Disc-structure space of compact smooth manifolds.

We establish an $h$-principle for exact Lagrangian embeddings with concave Legendrian boundary. We prove, in particular, that in the complement of the unit ball $B$ in the standard symplectic $\R^{2n}, 2n\geq 6$, there exists an embedded Lagrangian $n$-disc transversely attached to $B$ along its Legendrian boundary.

It is well-known that all 2-knots are slice. Are all 2-links slice? This is an outstanding open question. In this paper we prove the following: For any 2-component 2-link (J,K)in the 4-sphere which bounds the 5-ball B^5, there is an embedded disc 2-disc D^2_J (respectively, D^2_K) in B^5 with the following properties: …

Unified quantum invariants via intersections of embedded Lagrangians.

Ancient curve shortening flow in a disc with mixed boundary conditions is solved.

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb R^3$. In this note we show that distortion minimisers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $π/2$ is…

We establish existence and regularity results for normal Coulomb frames in the normal bundle of two-dimensional surfaces of disc-type embedded in Euclidean spaces of higher dimensions.

The affine diffeomorphism group $\mathrm{Aff}(S,q)$ of a half-translation surface $(S,q)$ comprise the self-diffeomorphisms with constant differential away from the singularities. This group coincides with the stabiliser of the associated Teichmüller disc under the action of the mapping class group on Teichmüller space…

We study a certain class of embedded two-foams that arise from gluing discs into ribbon torus knots along nonintersecting torus meridians. We exhibit several equivalent diagrammatic formalisms for these objects and identify several of their invariants, including a unique prime decomposition.

Constructs minimal surfaces near the boundary of a ball.

Study compares constrained and decoupled moduli spaces of manifolds with particles and discs.

We prove that the embedding of the quaternionic hyperbolic disc $H^1_\mathbb{H}$ into quaternionic hyperbolic $n$-space $H^n_\mathbb{H}$ is tight and thereby obtain the value of the Gromov norm of the quaternionic Kähler class.

An extremal $k$-packing is a collection of $k$ mutually disjoint metric discs, embedded in a surface, whose radius is maximal for the given topology. We study compact non-orientable surfaces of genus $g\ge 3$ containing extremal $k$-packings.

We construct and analyze minimal disc stackings with bounds on their Morse index.

Sharp estimate on harmonic maps at conformal points in balls.

We obtain global extensions of the celebrated Nash-Kuiper theorem for $C^{1,θ}$ isometric immersions of compact manifolds with optimal Hölder exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent $θ<1…

Study models Indian stock market using hyperbolic geometry for market stability and volatility analysis.

Paper equates torsions on wedge singularities.

Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative integer $m$ we find a pair of 2-knots in the 4-sphere whose stabilization…

Smoothly isotopic 3-discs in 4-sphere become identical after 5-dimensional push.

The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.

We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmüller space, if the quasicircle is sufficiently close to being the bou…