In this paper, we prove that the Schrödinger map flows from ${\mathbb{R}}^d$ with $d\ge 3$ to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. This is a companion work of our previous paper [21] where the energy critical case $d=2$ was solved. In the first part of this paper, for heat flows from ${\mathbb{R}}^d$ ($d\ge 3$) to Riemannian manifolds with small data in critical Sobolev spaces, we prove the decay estimates of moving frame dependent quantities in the caloric gauge setting, which is of independent interest and may be applied to other problems. In the second part, with a key bootstrap-iteration scheme in our previous work [21], we apply these decay estimates to the study of Schrödinger map flows by choosing caloric gauge. This work with our previous work solves the open problem raised by Tataru.

在本文中，我们证明了薛定ding图来自 ${\mathbb{[R}}^d$ 和 $d\ge 3$在关键的Sobolev空间中以小的初始数据来压缩Kähler流形是全球性的。这是我们先前论文[21]的伴随工作，其中能源关键案例$d=\mathrm{2个}$解决了。在本文的第一部分中，${\mathbb{[R}}^d$ （$d\ge 3$）对于临界Sobolev空间中具有小数据的黎曼流形，我们证明了在热量规设置中与运动框架相关的量的衰减估计，这是独立关注的，并且可能适用于其他问题。在第二部分中，我们在先前的工作中采用了关键的自举迭代方案[21]，我们通过选择热量计将这些衰减估计应用于Schrödinger地图流的研究。这项工作与我们以前的工作一起解决了Tataru提出的开放性问题。