arXiv research

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…

Counterexamples show no simple generalization of Hasimoto transform for higher-dimensional Euler fluids.

Paper formulates governing equations for membrane O surfaces.

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 …

New multi-spiral approach improves packaging of thick membranes.

New theory shows how membranes can break symmetry.

This review reports some theoretical results on the Geometry of membranes. The governing equations to describe equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are derived from the variation of free energies of these structures. Some analytic solutions to these e…

Study the stability of membranes using Helfrich energy and second variation formula.

Recent theoretical advances in elasticity of membranes following Helfrich's famous spontaneous curvature model are summarized in this review. The governing equations describing equilibrium configurations of lipid vesicles, lipid membranes with free edges, and chiral lipid membranes are presented. Several analytic solut…

Proves $C^{1,1}$ regularity for multiple membrane solutions.

The purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status in theoretical and experimental studies on the shapes of bio-membranes, a geometric scheme is proposed to discuss the shape equation of closed…

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…

Periodic surfaces have a limited number of bending modes, equal to their membrane modes.

The paper introduces vortex nerve complexes and new Betti numbers in CW spaces.

This review reports some key results in theoretical investigations on configurations of lipid membranes and presents several challenges in this field which involve (i) exact solutions to the shape equation of lipid vesicles; (ii) exact solutions to the governing equations of open lipid membranes; (iii) neck condition o…

Study vortex loops as coadjoint orbits of diffeomorphisms.

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

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

The article extends vortex filament theory to Hermitian reductive Lie algebras.

Study connects hyperbolic geometry to membrane shapes.

Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane surface is usually assumed to be at least $C^1$ across phase boundaries. We consi…

We develop theory and computational methods to investigate particle inclusions embedded within curved lipid bilayer membranes. We consider the case of spherical lipid vesicles where inclusion particles are coupled through (i) intramembrane hydrodynamics, (ii) traction stresses with the external and trapped solvent flui…

New minimal surfaces found from vortex crystals.

We construct membrane homology groups $\h(M)$ associated with each compact connected oriented smooth manifold, and show that $\h(M)$ is matrix graded algebra.

We define a topological quantum membrane theory on a seven dimensional manifold of $G_2$ holonomy. We describe in detail the path integral evaluation for membrane geometries given by circle bundles over Riemann surfaces. We show that when the target space is $CY_3\times S^1$ quantum amplitudes of non-local observables …

We construct a gauge fixed action for topological membranes on $G_2$-manifold such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds. Moreover on $M\times S^1$ the theory naturally reduces…

Study of symplectic manifolds degenerating into singular spaces.

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

Solved Demailly systems for Vortex bundles on manifolds.

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 …

We investigate geometric aspects of double field theory (DFT) and its formulation as a doubled membrane sigma-model. Starting from the standard Courant algebroid over the phase space of an open membrane, we determine a splitting and a projection to a subbundle that sends the Courant algebroid operations to the correspo…

Alternative construction of quasi-Fuchsian flows using vortex equations.

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

Proves Payne conjecture for buckling and membrane eigenvalues.

We consider a diffuse interface approximation for the lipid phases of rotationally symmetric two-phase bilayer membranes and rigorously derive its $Γ$-limit. In particular, we prove that limit vesicles are $C^1$ across interfaces, which justifies a regularity assumption that is widely made in formal asymptotic and nume…

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…

Analytic formulae for point vortex dynamics on surfaces with symmetry.

Theory of point vortices extended to closed surfaces.

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

We give a definition of higher dimensional iterated integrals based on integration over membranes. We prove basic properties of this definition and formulate a conjecture which extends Chen's de Rham Theorem for iterated integrals to the membrane case.

Topologically protected vortex knots and links are proposed and proven.

For a membrane in the plane the multiplicity of the $k$-th eigenvalue is known to be not greater than $2k-1$. Here we prove that it is actually not greater than $2k-3$, for $k\ge 3$.

A surface functional theory for p-dimensional extended objects, the p-branes, was proposed in previous papers. The field equations for toroidal p-branes was exactly solved in $d=p+2$ dimensions, yielding equally spaced mass-squared spectrum with massless states. In this paper, we obtain the asymptotic distribution of m…

Researchers create topologically protected knots in a realizable system.

Transforming cylindrical packings into bicontinuous surfaces.

Proves existence and regularity of spherical minimizers for lipid membrane energy.

This paper mainly aims to establish the well-posedness on time interval $[0,\varepsilon^{-\frac{1}{2}}T]$ of the classical initial problem for the bosonic membrane in the light cone gauge. Here $\varepsilon$ is the small parameter measures the nonlinear effects. In geometric, the bosonic membrane are timelike submanifo…

Bayesian modeling predicts hydroxide ion conductivity in polymer membranes.