Related papers: Moving null curves and integrability
A connection is established between the soliton equations and curves moving in a three dimensional space $V_{3}$. The sign of the self-interacting terms of the soliton equations are related to the signature of $V_{3}$. It is shown that…
We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also,…
We develop a novel approach to the Brill-Noether theory of curves endowed with a degree k cover of the projective line via Bridgeland stability conditions on elliptic K3 surfaces. We first develop the Brill-Noether theory on elliptic K3…
In this paper we consider the equiform motion of a helix in Euclidean space $\mathbf{E}^7$. We study and analyze the corresponding kinematic three dimensional surface under the hypothesis that its scalar curvature $\mathbf{K}$ is constant.…
In this work, we consider 2-surfaces in ${\mathbb R}^3$ arising from the modified Korteweg de Vries (mKdV) equation. We give a method for constructing the position vector of the mKdV surface explicitly for a given solution of the mKdV…
We investigate the geometry of a particular class of null surfaces in space-time called vacuum Non-Expanding Horizons (NEHs). Using the spin-coefficient equation, we provide a complete description of the horizon geometry, as well as fixing…
We investigate complete minimal hypersurfaces in the Euclidean space $% \ {R}^{4}$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\to {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature…
In this paper, the Kelvin wave and knot dynamics are studied on three dimensional smoothly deformed entangled vortex-membranes in five dimensional space. Owing to the existence of local Lorentz invariance and diffeomorphism invariance, in…
The nature of the evolution of the magnetic field, and of current sheet formation, at three-dimensional (3D) magnetic null points is investigated. A kinematic example is presented which demonstrates that for certain evolutions of a 3D null…
We consider the dynamics of moving curves in three-dimensional Minkowski space $R_1^{3}$ and deduce the evolution equations for the curvature and torsion of the curve. Next by mapping a continuous SO(2,1) Heisenberg spin chain on the space…
We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the singular Bj\"orling problem for such surfaces, which is stated as follows: given a real…
In this paper, we investigate the caustic point developed by outgoing null hypersurfaces and the behavior of null curves. We analyze three cases: Future, past, and zero caustic point terminology for outgoing null hypersurfaces in the…
We use the isotropic projection of Laguerre geometry in order to establish a correspondence between plane curves and null curves in the Minkowski $3$-space. We describe the geometry of null curves (Cartan frame, pseudo-arc parameter,…
Morphodynamic equations governing the behaviour of active nematic fluids on deformable curved surfaces are constructed in the large deformation limit. Emphasis is placed on the formulation of objective rates that account for normal…
The problem of existence of spacelike hypersurfaces with constant mean curvature in asymptotically flat spacetimes is considered for a class of asymptotically Schwarzschild spacetimes satisfying an interior condition. Using a barrier…
One of the central difficulties of settling the $L^2$-bounded curvature conjecture for the Einstein -Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to…
We study the totally null surfaces of the neutral Kaehler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual ($\alpha$-planes) or anti-self-dual ($\beta$-planes) and so we consider…
A certain class of surface motions, including those of a relativistic membrane minimizing the 3-dimensional volume swept out in Minkowski-space, is shown to be equivalent to 3-dimensional steady-state irrotational inviscid isentropic…
Context: Coronal magnetic null points have been implicated as possible locations for localised heating events in 2D models. We investigate this possibility about fully 3D null points. Aims: We investigate the nature of the fast…
We apply our recent formalism establishing new connections between the geometry of moving space curves and soliton equations, to the nonlinear Schr\"{o}dinger equation (NLS). We show that any given solution of the NLS gets associated with…