论文标题
$ d \ geq 2 $ dimensions中的小数据的偏斜平均曲率流量的本地稳定性
Local well-posedness of the Skew mean curvature flow for small data in $d\geq 2$ dimensions
论文作者
论文摘要
偏斜平均曲率流是$ \ Mathbb {r}^{d+2} $嵌入$ d $ dimensional歧管的进化方程（或更一般而言，在Riemannian歧管中）。它可以看作是平均曲率流的Schrödinger类似物，也可以看作是Schrödinger图方程的准线性版本。在较早的论文中，作者引入了该问题的谐波/库仑仪表，并用它证明了小数据在尺寸中局部良好的数据$ d \ geq 4 $。在本文中，我们证明了小数据在低规范性Sobolev空间中的本地适合度，用于偏度曲率流量$ d \ geq 2 $。这是通过引入新方程式的新的热量表公式来实现的，该方程式在低维度中变得更健壮。
The skew mean curvature flow is an evolution equation for $d$ dimensional manifolds embedded in $\mathbb{R}^{d+2}$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions $d \geq 4$. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $d\geq 2$. This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.
