Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global Schr\"odinger map flows to K\"ahler manifolds with small data in
  critical Sobolev spaces: High dimensions   In this paper, we prove that the Schr\"odinger map flows from $\Bbb R^d$ 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 [23]
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. In the second part,
with a key bootstrap-iteration scheme in our previous work [23], we apply these
decay estimates to the study of Schr\"odinger map flows by choosing caloric
gauge. This work with our previous work solves the open problem raised by
Tataru.

33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con