Narrow Results

In this paper, by using the $G_2$-structure on Im$(𝕆)\cong {ℝ}^7$ from the octonions $𝕆$, the $G_2$-binormal motion of curves $\gamma (t,s)$ in ${ℝ}^7$ associated to the almost complex structure on ${𝕊}^6$ is studied. The motion is proved to be equivalent to Schrödinger flows from ${ℝ}^1$ to ${𝕊}^6$, and also to a nonlinear Schrödinger-type system (NLSS) in three unknown complex functions that generalizes the famous correspondence between the binormal motion of curves in ${ℝ}^3$ and the focusing nonlinear Schrödinger (NLS) equation. Some related geometric properties of the surface $\mathrm{\Sigma }$ in Im$(𝕆)$ swept by $\gamma (t,s)$ are determined.

The skew mean curvature flow (SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space ${ℝ}^4$. A Sobolev-type embedding theorem for the second fundamental forms of two-dimensional surfaces is also proved, which might be of independent interest.

The definition of Schrödinger flow is proposed. It is indicated that the flow of ferromagnetic chain is actually Schrödinger flow of maps into S², and that there exists a unique local smooth solution for the initial value problem of one-dimensional Schrödinger flow of maps into Kähler manifolds. In case the targets are Kähler manifolds with constant curvature, it is proved that one-dimensional Schrödinger flow admits a unique global smooth solution.

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schrödinger flow for maps from a compact Riemannian manifold into a complete Kähler manifold, or from a Euclidean space R^m into a compact Kähler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.

In this note, we prove that the Schrödinger flow of maps from a closed Riemann surface into a compact irreducible Hermitian symmetic space admits a global weak solution. Also, we show the existence of weak solutions to the initial value problem of Heisenberg model with Lie algebra values, which is closely related to the Schrödinger flow on compact Hermitian symmetric spaces.