Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: High dimensions

Abstract

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 [23] 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 [23], 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.

slightly enlarged, submitted