Vortex Dynamics in Any Dimension
Staley, Phillip Lee
MetadataShow full item record
We adapt the general method of moving frames and time-dependent submanifold geometry to investigate the <i>skew-mean curvature flow</i> of a 2-dimensional surface in R4; this is a natural generalization of the <i>vortex filament equation</i> (VFE) for the motion a vortex membrane in 4-dimensional space. (Haller and Vizman, Khesin, Shashinkanth.) We propose a geometric framework (the <i>Hasimoto bundle</i>) in which the connection between the VFE and the nonlinear Schrödinger equation (through the well-known <i>Hasimoto transformation</i>) becomes transparent, and that easily generalizes to the general codimension-2 case. Such framework reveals that the main obstruction to the existence of a parallel frame and to defining a natural Hasimoto transformation is the non-zero curvature of the normal bundle.