arXiv research

New metrics help predict Brownian motion on surfaces and higher dimensions.

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve t…

We demonstrate that the five vortex equations recently introduced by Manton ariseas symmetry reductions of the anti-self-dual Yang--Mills equations in four dimensions. In particular the Jackiw--Pi vortex and the Ambjørn--Olesen vortex correspond to the gauge group $SU(1, 1)$, and respectively the Euclidean or the $SU(2…

Alternative construction of quasi-Fuchsian flows using vortex equations.

Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.

Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.

The paper proves estimates for vortex-type equations on compact Riemann surfaces.

We introduce a notion of Gieseker stability for a filtered holomorphic vector bundle $F$ over a projective manifold. We relate it to an analytic condition in terms of hermitian metrics on $F$ coming from a construction of the Geometric Invariant Theory (G.I.T). These metrics are balanced in the sense of S.K. Donaldson.…

In this note we quantize the usual symplectic (Kähler) form on the vortex moduli space by modifying the Quillen metric of the Quillen determinant line bundle.

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 …

It is known that given a stable holomorphic pair $(E ,φ)$, where $E$ is a holomorphic vector bundle on a compact Kähler manifold $X$ and $φ$ is a holomorphic section of $E$, the vector bundle $E$ admits a Hermitian metric solving the vortex equation. We generalize this to pairs $(\E ,φ)$, where $\E$ is a reflexive shea…

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 …

Paper extends Kobayashi-Hitchin correspondence to non-Kähler manifolds.

The two-loop renormalization group flow is studied via the induced bracket flow on 3D unimodular Lie groups. A number of steady solitons are found. Some of these steady solitons come from maximally symmetric metrics that are steady, shrinking, or expanding solitons under Ricci flow, while others are not obviously relat…

Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show that a pair (E, φ), consisting of a flat vector bundle E over M and a flat nonzero section φ of E, admits a solution to the vortex equation if and only if it is polystable. To prove this, we adapt the dimensional redu…

At critical coupling, the interactions of Ginzburg-Landau vortices are determined by the metric on the moduli space of static solutions. The asymptotic form of the metric for two well separated vortices is shown here to be expressible in terms of a Bessel function. A straightforward extension gives the metric for N vor…

In this work we consider the gravitating vortex equations. These equations couple a metric over a compact Riemann surface with a hermitian metric over a holomorphic line bundle equipped with a fixed global section --- the Higgs field ---, and have a symplectic interpretation as moment-map equations. As a particular cas…

We consider the self-dual vortex equations on a positive line bundle L --> M over a compact Kaehler manifold of arbitrary dimension. When M is simply connected, the moduli space of vortex solutions is a projective space. When M is an abelian variety, the moduli space is the projectivization of the Fourier-Mukai transfo…

We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e. singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximat…

We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton struc…

We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces. In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kaehler class of the L^2-metric. As an application we compute the total volume of the moduli …

New explicit Calabi-Yau metrics and Kähler-Ricci solitons found on complex n-space.

We prove that the isoperimetric inequality is satisfied in the cigar steady soliton and in the Bryant steady soliton. Since both of them are Riemannian manifolds with warped product metric, we utilize the result of Guan-Li-Wang to get our conclusion. For the sake of the soliton structure, we believe that the geometric …

Study vortex loops as coadjoint orbits of diffeomorphisms.

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…

Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.

Study proves rigidity of certain gradient steady Ricci solitons with harmonic Weyl curvature.

New minimal surfaces found from vortex crystals.

The paper studies point vortex dynamics on specific Kähler twistor spaces.

Researchers found infinitely many non-collapsed steady Ricci solitons on complex line bundles.

Solved Demailly systems for Vortex bundles on manifolds.

Study on Kähler-Ricci solitons on toric manifolds, proving rigidity.

We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological characterization of geodesible flows in the volume-preserving case. As an application, we…

Uniform convergence of metrics on vortex moduli space in Bradlow limit.

We obtain a Hitchin-Kobayashi-type correspondence for symplectic vortex equations, with the target a Kahler cone over a compact Sasakian manifold. We show that the correspondence reduces to studying the existence and uniqueness of Kazdan-Warner equations. Using this, we construct a map between the moduli space of solut…

In this paper we have proved that a compact Riemannian manifold does not admit a metric with positive scalar curvature if there exists a real valued function in this manifold which is strictly positive along a geodesic ray satisfying expanding or steady Yamabe soliton. We have also deduced a relation between scalar cur…

Theory of point vortices extended to closed surfaces.

Study how large-scale flows align small-scale vortices in 3D Euler equations.

Topologically protected vortex knots and links are proposed and proven.

Constructs complete metrics and solitons on complex vector bundles.

Researchers create topologically protected knots in a realizable system.

The paper explores the geometric properties of fluid flows and their symmetries.

Study on asymptotic behavior of Taub-NUT type solitons and construction of new ALF Calabi-Yau metrics.

The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton $N^2\times \mathbb{R}$, the product of Hamilton's cigar soliton $N^2$ and the real line $\mathbb{R}$ with the product metric. R. Hamilton has conjectured that there should exist a family of colla…

We consider the vortex equations for a U(n) gauge field coupled to a Higgs field with values on the n times n square matrices. It is known that when these equations are defined on a compact Riemann surface, their moduli space of solutions is closely related to a moduli space of tau-stable holomorphic n-pairs on that su…

With a f-left-invariant Riemannian metric on a Lie group $G$, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor $f$. In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Rie…

The gauged sigma model with target $\mathbb{P}^1$, defined on a Riemann surface $Σ$, supports static solutions in which $k_+$ vortices coexist in stable equilibrium with $k_-$ antivortices. Their moduli space is a noncompact complex manifold $M_{(k_+,k_-)}(Σ)$ of dimension $k_++k_-$ which inherits a natural Kähler metr…