arXiv research

Finite-type k-surfaces in hyperbolic space have finite genus and cusp-like ends.

Introduces Möbius structures and hyperbolic ends for $k$-surfaces in hyperbolic space.

Study on $k$-surfaces in negatively curved 3-manifolds, focusing on energy and entropy.

We define the ``volume'' contained by pointed $k$-surfaces, first studied by the author in [9], and we show that this volume is always finite. Likewise, we show that the surface area of a pointed $k$-surface is always finite.

Study classifies special metrics on specific surfaces.

The main result is the construction of ergodic transversal measures of full support on the space of all k-surfaces of a compact hyperbolic 3-manifold. This space is a laminated space, each of its leaf being identified with a "complete" k-surface, i.e. a surface of constant (extrinsic) curvature k, where k belongs to ]0…

We prove that every complete connected immersed surface with positive extrinsic curvature $K$ in $H^2\times R$ must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study the behavior of the end. Then, we focus our attention on surfaces with positive constant extrinsic curvature ($K-$s…

We study the basic structure of a HCMU metric in a K-Surface with prescribed singularities. When the underlying smooth surface is $S^2$, we prove the necessary condition given in [1] for the existence of HCMU metric is also sufficient.

We prove that any weakly acausal curve $Γ$ in the boundary of Anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces, one of which is past-convex and the other future-convex, for every $K\in(-\infty,-1)$. The curve $Γ$ is the graph of a quasisymmetric homeomorphism of the circle if and only…

Study counts surface subgroups in curved 3D manifolds.

We describe the "hyperbolic" properties of a riemann surface lamination M canonically associated to every compact three manifolds of curvature less than 1. More precisely, if the geodesic flow is the phase space attached to an ordinary differential equation, our space M is the "phase space" attached to a certain ellipt…

A well-known question in classical differential geometry and geometric analysis asks for a description of possible boundaries of $K$-surfaces, which are smooth, compact hypersurfaces in $\mathbb{R}^d$ having constant Gauss curvature equal to $K \geq 0$. This question generated a considerable amount of remarkable result…

Following on from ``Hyperbolic Plateau problems'' (by the same author), we provide a complete geometric description of solutions to the Plateau problem $(S,φ)$ when $S$ is a compact Riemann surface with a finite number of points removed.

A numerical scheme is developed for solution of the Goursat problem for a class of nonlinear hyperbolic systems with an arbitrary number of independent variables. Convergence results are proved for this difference scheme. These results are applied to hyperbolic systems of differential-geometric origin, like the sine-Go…

Simple constructions of semi-discrete and discrete surfaces using Jacobi elliptic functions.

Category theory generalizes finite type invariants using diagrams systems.

Quaternionic reformulation simplifies surface curvature theory.

Submanifolds of coordinate finite-type were introduced in HV1. A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of Δ. In the present study we consider coordinate finite-type surfaces in E^4. We give necessary and sufficient conditions for g…

Defines finite type Multivalued Shape using hyperspaces.

Finite type curves are asymptotic lines of plane fields, with examples.

New invariant fully describes finite type invariants of knots in homology 3-spheres.

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite t…

Compact manifolds with boundary have finite type mapping class groups.

The paper defines biquandles for groups and constructs a homomorphism.

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low de…

Finite type invariants separate PL links in 3D space.

Cubic complexes appear in the theory of finite type invariants so often that one can ascribe them to basic notions of the theory. In this paper we begin the exposition of finite type invariants from the `cubic' point of view. Finite type invariants of knots and homology 3-spheres fit perfectly into this conception. In …

We observe that most known results of the form "v is not a finite-type invariant" follow from two basic theorems. Among those invariants which are not of finite type, we discuss examples which are "ft-independent" and examples which are not. We introduce (n,q)-finite invariants, which are generalizations of finite-type…

Finite-type surfaces have a topological Hopf property.

We enhance the biquandle counting invariant using elements of truncated biquandle-labeled Polyak algebras. These finite type enhancements reduce to the finite type enhancements defined by Goussarov, Polyak and Viro for the trivial biquandle of one element and determine (but are not determined by) the biquandle counting…

A theory of finite type invariants for arbitrary compact oriented 3-manifolds is proposed, and illustrated through many examples arising from both classical and quantum topology. The theory is seen to be highly non-trivial even for manifolds with large first betti number, encompassing much of the complexity of Ohtsuki'…

Delta finite-type invariants are defined analogously to finite-type invariants, using delta moves instead of crossing changes. We show that they are closely related to the lower central series of the commutator subgroup of the pure braid group.

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject had been collected in author's book [Total mean curvature and sub manifolds of finite type, World Scientific, NJ, 1984]. A list of ten open problems and three conjectures on submanifolds of finite type was…

In this paper, we define finite type invariants for cyclic equivalence classes of nanophrases and construct the universal ones. Also, we identify the universal finite type invariant of degree 1 essentially with the linking matrix. It is known that extended Arnold's basic invariants to signed words are finite type invar…

A smooth four manifold is of finite type $r$ if its Donaldson invariant satisfies D((x^2-4)^r)=0. We prove that every simply connected manifold is of finite type by using the structure of Donaldson invariants in the presence of immersed spheres. More precisely we prove that if a manifold X contains an immersed sphere w…

This is an overview article on finite type invariants, written for the Encyclopedia of Mathematical Physics

This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.

In this paper, we give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to …

Study on compact and finite-type support in mapping class group homology.

In this paper, it is shown that there are no nonconstant Goussarov-Polyak-Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem which states none of the Birman coefficients of the Jones-Kauffman polynomial are of GPV finite type.

We construct a hyperbolic three-manifold with trivial finite type invariants up to a given degree.

Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the p…

Study proves all free boundary CMC annuli are of finite type.

We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects previous errors of…

Artin groups of finite type are not as well understood as braid groups. This is due to the additional geometric properties of braid groups coming from their close connection to mapping class groups. For each Artin group of finite type, we construct a space (simplicial complex) analogous to Teichmueller space that satis…

Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type inva…

An explicit polynomial in the linking numbers $l_{ij}$ and Milnor's triple linking numbers $μ(rst)$ on six component links is shown to be a well-defined finite type link-homotopy invariant. This solves a problem raised by B. Mellor and D. Thurston. An extension of our construction also produces a finite type link invar…

Efficient algorithm computes knot invariants quickly.