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

Ze Li

Published 2019-03-13Version 1

In this paper, we prove that the Schr\"odinger map flows from $R^d$ with $d\ge 3$ to compact K\"ahler manifolds with small initial data in critical Sobolev spaces are global. This is a companion work of our previous paper where the energy critical case $d=2$ was solved. In the first part of this paper, for heat flows from $\Bbb R^d$ ($d\ge 3$) 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.