arXiv research

Proves inequality for 1-dimensional cycles.

Proves the Weyl law for 1-cycles in manifolds.

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold M^n is bounded from above by (n+2)!d/3,…

This article introduces proximal planar vortex 1-cycles, resembling the structure of vortex atoms introduced by William Thomson (Lord Kelvin) in 1867 and recent work on the proximity of sets that overlap either spatially or descriptively. Vortex cycles resemble Thomson's model of a vortex atom, inspired by P.G. Tait's …

This article introduces planar shape signatures derived from homology nerves, which are intersecting 1-cycles in a collection of homology groups endowed with a proximal relator (set of nearness relations) that includes a descriptive proximity. A 1-cycle is a closed, connected path with a zero boundary in a simplicial c…

Study 2-complexes' homology properties and torsion growth.

We prove that given a three manifold with an arbitrary metric $(M^3, g)$ of positive Ricci curvature, there exists a sweepout of $M$ by surfaces of genus $\leq 3$ and areas bounded by $C vol(M^3, g)^{2/3}$. We use this result to construct a sweepout of $M$ by 1-cycles of length at most $C vol(M^3, g)^{1/3}$. The sweepo…

We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines…

The paper shows equidistribution of geodesics and nets on manifolds.

New Vassiliev invariant of order three derived from Fox-Hatcher cycles.

Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M,g)=1 and every 1-cycle dividing M into two regions of equal area has length >C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Ste…

Let $M$ be a closed Riemannian surface of genus $g$. We construct a family of 1-cycles on $M$ that represents a non-trivial element of the k'th homology group of the space of cycles and such that the mass of each cycle is bounded above by $C \max\{\sqrt{k}, \sqrt{g}\} \sqrt{Area(M)}$. This result is optimal up to a mul…

New parities defined on virtual knots linked to crossing indices.

We give a new proof of the Alexander-Wermer Theorem that characterizes the oriented curves in C^n which bound positive holomorphic chains, in terms of the linking numbers of the curve with algebraic cycles in the complement. In fact, we establish a slightly stronger version which applies to a wider class of boundary 1-…

Derives Weyl law for volume spectrum using parametric inequalities.

We discuss the explicit formulation of the transcendental constraints defining spectral curves of SU(2) BPS monopoles in the twistor approach of Hitchin, following Ercolani and Sinha. We obtain an improved version of the Ercolani-Sinha constraints, and show that the Corrigan-Goddard conditions for constructing monopole…

We consider the classical problem of a position of n-dimensional manifold M in R^{n+2}. We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting M to R^{n+2}. In particular, we show that for any fixed quandle, quandle coloring, and shadow quan…

We prove that the mirror map is trivial for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author. In other words, the natural coordinate in a canonical Calabi-Yau family is a canonical coordinate in the sense of Hodge theory. This implies that the higher weight periods direct…

The projection of a compact oriented submanifold M^{n-1} in R^{n+1} on a hyperplane P^{n} can fail to bound any region in P. We call this ``projecting to zero.'' Example: The equatorial S^1 in S^2 projects to zero in any plane containing the x_3-axis. Using currents to make this precise, we show: A lipschitz (homology)…

A formula calculates the Euler class of foliations using dual graphs.

The paper proves existence and partial regularity for Legendrian area-minimizing currents.

A method for constructing explicit Calabi-Yau metrics in six dimensions in terms of an initial hyperkahler structure is presented. The equations to solve are non linear in general, but become linear when the objects describing the metric depend on only one complex coordinate of the hyperkahler 4-dimensional space and i…

We prove the existence of Lagrangian fillings for $D_n$-type Legendrian links.

This article introduces vortex nerve complexes in CW (Closure finite Weak) topological spaces, which first appeared in works by P. Alexandroff, H. Hopf and J.H.C. Whitehead during the 1930s. A vortex nerve is a CW complex containing one or more intersecting path-connected cycles. Each vortex nerve has its own distincti…

Continuous sweepouts cover manifolds with bounded curve lengths.

This paper initiates the study of topological arbiters, a concept rooted in Poincare-Lefschetz duality. Given an n-dimensional manifold W, a topological arbiter associates a value 0 or 1 to codimension zero submanifolds of W, subject to natural topological and duality axioms. For example, there is a unique arbiter on $…

We consider the stable norm associated to a discrete, torsionless abelian group of isometries $Γ\cong \mathbb{Z}^n$ of a geodesic space $(X,d)$. We show that the difference between the stable norm $\| \;\, \|_{st}$ and the distance $d$ is bounded by a constant only depending on the rank $n$ and on upper bounds for the …

Little is known about the global structure of the basins of attraction of Newton's method in two or more complex variables. We make the first steps by focusing on the specific Newton mapping to solve for the common roots of $P(x,y) = x(1-x)$ and $Q(x,y) = y^2+Bxy-y$. There are invariant circles $S_0$ and $S_1$ within t…

This article introduces planar ribbons, Vergili ribbon complexes and ribbon nerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topological spaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closure of a pair of nesting, non-concentric filled cycles that includes the boundary but does not …

A new geometric framework resolves singularities in anomalous transport.