Vortex solutions on flat surfaces map to harmonic spinors on Nappi-Witten space.
problem Constructing Abelian magnetic zero-modes on flat spacetime.
method Establishing a correspondence between vortex equations and harmonic spinors on the Nappi-Witten space.
result Explicit solutions of a twisted Dirac equation induce harmonic spinors on Minkowski space.
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 …
Paper shows existence of vortex solutions with specific decay properties.
problem Existence of solutions to Seiberg-Witten equations with specific decay properties.
method Dimensional reduction of Seiberg-Witten equations on the plane.
result Contains both exponentially decayed and polynomial growth solutions.
Symplectic vortex equations link Sasakian manifolds to Kahler cones.
problem Existence and uniqueness of symplectic vortex solutions.
method Hitchin-Kobayashi correspondence and Kazdan-Warner equations.
result Construction of a map between vortex solutions and effective divisors.
Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.
problem Existence of solutions to doubly-coupled vortex equations on Riemann surfaces.
method Introduced doubly-coupled vortex equations and used Higgs bundle theory.
result Existence of solutions to vortex equations is equivalent to Higgs bundle stability.
Solved Demailly systems for Vortex bundles on manifolds.
problem Equivalence of Hartshorne ampleness and Griffiths positivity for vector bundles.
method Applied the continuity method to prove smooth solutions for the Vortex bundle.
result Smooth solutions exist for the proposed Demailly systems.
Extending work of Caffarelli-Yang and Tarantello, we present a variational existence proof for two-vortex solutions of the periodic Chern-Simons Higgs model and analyze the asymptotic behavior of these solutions as the parameter coupling the gauge field with the scalar field tends to 0.
Study of vortex equations' limits on growing surfaces.
problem Understanding vortex equations' behavior as surface volume increases.
method Prior estimates for Kazdan-Warner equations, relating to previous work.
result Solutions converge smoothly away from finitely many points.
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…
For the class of quasi-periodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction and the geometry of the evolving curves. We give a complete description of genus one solutions, including geometrically interesting special cases such a…
Study finds obstacles to solutions for specific equations on compact surfaces.
problem Existence of solutions to self-dual equations on compact surfaces.
method Depends on Higgs field zeroes and vortex number.
result Infinitely many Higgs fields for which solutions cannot exist.
We study the topology of quasiperiodic solutions of the vortex filament equation in a neighborhood of multiply covered circles. We construct these solutions by means of a sequence of isoperiodic deformations, at each step of which a real double point is "unpinched" to produce a new pair of branch points and therefore a…
Giving a new form of the vortex mode equation by a proper change of parameter, our aim is to analyze the point and contact symmetries of the new equation. Fundamental invariants and a form of general solutions of point transformations along with some specific examples are also derived.
Study vortex flows on Riemann surfaces, proving dominated splitting and Anosov properties.
problem Investigate flow properties on Riemann surfaces.
method Associate flow to vortex equations, investigate properties of flow.
result Show that flow always admits a dominated splitting and identify special cases of Anosov flow.
We consider nonlinear gauged sigma-models with Kahler domain and target. For a special choice of potential these models admit Bogomolny (or self-duality) equations -- the so-called vortex equations. We find the moduli space and energy spectrum of the solutions of these equations when the gauge group is a torus T^n, the…
Stable solutions to Yang-Mills-Higgs equations on spheres and tori identified.
problem Stable solutions to abelian Yang-Mills-Higgs equations on S2 and T2. method Reduction to vortex equations and application of Bourguignon-Lawson's method for stable SU(2) Yang-Mills connections. result Stable solutions to abelian Yang-Mills-Higgs equations on S2 and T2 are identified as satisfying vortex equations. Counterexamples show no simple generalization of Hasimoto transform for higher-dimensional Euler fluids.
problem No straightforward generalization of Hasimoto transform for higher-dimensional Euler fluids.
method Derivation of evolution equations for mean curvature and torsion form for membranes.
result Existence of counterexamples implies no simple generalization of Hasimoto transform.
Study connects vortices in abelian Higgs models to discrete conformal maps.
problem Understanding vortices in abelian Higgs models.
method Relating vortices to discrete conformal maps via curvature and volume form.
result Constructing discrete vortex solutions using discrete conformal theory.
The paper introduces vortex cycles and nerves, inspired by Thomson's vortex atoms.
problem Understanding vortex structures and their homology.
method Introducing and analyzing non-concentric, nesting vortex cycles and nerves.
result Whitehead CW topology and Leader uniform topology outcomes of vortex cycles.
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…
Study on vortex sheet formation in Abelian gauge theories.
problem Understanding vortex sheet formation in Abelian gauge theories.
method Inspired by Allard's regularity theory, constructs approximate solutions and analyzes their perturbations.
result Establishes a geometric framework and regularity theory for the limiting defect set.
We consider the set of solutions to the rho-vortex equations over a Kahler surface and prove a Uhlenbeck compactness result, namely that a sequence of solutions with the same energy converge to the sum of a solution of smaller energy and deltas of Dirac.
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…
Manton's five vortex equations derived from Yang-Mills self-duality.
problem Deriving vortex equations from Yang-Mills theory.
method Symmetry reductions of anti-self-dual Yang-Mills equations.
result Vortex equations of different types can be superposed.
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…
The paper introduces vortex nerve complexes and new Betti numbers in CW spaces.
problem Understanding the structure and properties of CW complexes and their nerves.
method Introducing vortex nerve complexes and defining new Betti numbers for CW complexes.
result New Betti numbers (vortex Bvtex, vortex nerve BvNrv, shape Bsh) are introduced and studied. Neural Networks improve incompressible flow simulations without complex kernels.
problem Simulating incompressible flows accurately and efficiently.
method Integrates Neural Networks with Random Vortex Dynamics for incompressible Navier-Stokes equations.
result Strictly enforces physical properties like incompressibility and boundary conditions.
The paper proves estimates for vortex-type equations on compact Riemann surfaces.
problem Estimating vortex-type equations on compact Riemann surfaces.
method Proves \emph{a priori} estimates for vortex-type equations.
result Recover existing estimates for vortex bundle Monge-Ampère equation, prove existence and uniqueness for Calabi-Yang-Mills equations, and get estimates for J−vortex equation. Study vortex loops as coadjoint orbits of diffeomorphisms.
problem Understanding vortex loops in terms of coadjoint orbits.
method Analyzing vortex loops as coadjoint orbits of area-preserving diffeomorphisms.
result Vortex loops are coadjoint orbits of the diffeomorphism group.
The article extends vortex filament theory to Hermitian reductive Lie algebras.
problem Investigating vortex filaments in Hermitian reductive Lie algebras.
method Develops vortex models in a purely geometric way.
result Vortex models revert to known models in Hermitian symmetric Lie algebras.
Investigates J-equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of J-equation on holomorphic vector bundles. method Introduces and studies J-equation, provides algebraic and numerical criteria. result Provides an algebraic condition (asymptotic J-stability) and a numerical criterion for vortex bundles. The paper studies the geometry of vortex-antivortex pairs on Riemann surfaces.
problem The moduli space of BPS vortex-antivortex pairs on Riemann surfaces.
method Detailed study of the Kähler metric gL2 on the moduli space, using localization formulae and numerical analysis. result The moduli space M(k+,k−)(Σ) has finite volume and is geodesically incomplete for Σ=S2. Study heat flow for gauged holomorphic maps over Kähler manifolds.
problem Understanding the moduli space of gauged holomorphic maps.
method Gradient flow approach inspired by Atiyah and Bott, heat flow method.
result Global existence of smooth solutions of gradient flow equations.
The moduli space of solutions to the vortex equations on a Riemann surface are well known to have a symplectic (in fact Kähler) structure. We show this symplectic structure explictly and proceed to show a family of symplectic (in fact, Kähler) structures ΩΨ0 on the moduli space, parametrised by Ψ0, a section o…
New minimal surfaces found from vortex crystals.
problem Minimal surfaces and vortex crystals.
method Gluing helicoids into minimal surfaces.
result New minimal surfaces and vortex crystals discovered.
The paper studies point vortex dynamics on specific Kähler twistor spaces.
problem Point vortex dynamics on Kähler twistor spaces.
method Explicit expression for Green's function, Hamiltonian determination, momentum map calculation.
result Explicit equations of motion and momentum map for point vortex dynamics.
Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and Z-stability. result Equivalence between dHYM solutions and Z-stability for vortex type bundles. Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and Z-stability. result Equivalence between dHYM solutions and Z-stability for vortex type bundles. Alternative construction of quasi-Fuchsian flows using vortex equations.
problem Constructing quasi-Fuchsian flows.
method Using coupled vortex equations to construct quasi-Fuchsian flows as thermostats.
result Formulas for the marked length spectrum of quasi-Fuchsian flows.
We describe a topological field theory that studies the moduli space of solutions of the symplectic vortex equations. It contains as special cases the topological sigma-model and topological Yang-Mills over Kahler surfaces. The correlation functions of the theory are closely related to the recently introduced Hamiltoni…
Analytic formulae for point vortex dynamics on surfaces with symmetry.
problem Understanding point vortex behavior on surfaces with continuous symmetry.
method Derived analytic formulae for Green and Robin functions on surfaces with hydrodynamic Killing vector fields.
result Unified tool for detailed studies of point vortex dynamics and Euler-Arnold flows on surfaces with continuous symmetry.
Theory of point vortices extended to closed surfaces.
problem Extending point vortex dynamics to closed surfaces.
method Unified theory of point vortex dynamics on the plane, sphere, and closed surfaces.
result Comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity.
Study on unique vortex equations and their geometric implications.
problem Uniqueness of vortex equations involving entire functions.
method Analyzing entire functions on the complex plane and showing geometric applications.
result Uniqueness of harmonic maps and affine spherical immersions with polynomial differential constraints, but failure for non-polynomial entire functions.
Study how large-scale flows align small-scale vortices in 3D Euler equations.
problem Understanding how large-scale flows align small-scale vortices in 3D Euler equations.
method Constructing a Lagrangian coordinate to identify when the Lie bracket is zero and investigating the locality of the pressure term.
result Clarified conditions under which small-scale vortices are aligned by large-scale flows.
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…
Topologically protected vortex knots and links are proposed and proven.
problem Decaying of tangled vortex structures through local reconnections and strand crossings.
method Proposed and proven topologically protected vortex structures using non-Abelian topological vortices.
result Existence of topologically protected Q8-colored links and classification using the Q-invariant. Study on half spheres solves Nirenberg problem with complex blow-up analysis.
problem Prescribing scalar curvature on half spheres.
method Refined blow-up analysis of finite energy approximated solutions.
result Complex blow-up points and vortex problems reveal new connections.
Researchers create topologically protected knots in a realizable system.
problem Creating topologically protected vortex knots in experimentally realizable systems.
method Investigated non-Abelian vortices in tetrahedral order in spin-2 Bose--Einstein condensates and bent-core nematic liquid crystals.
result Discovered the first topologically protected knots in an experimentally realizable system.