4 1. INTRODUCTION

out to be necessary to control infinitely many Coulomb gauges, each of which corre-

spond to a distinct polarized plane wave solution to the usual (flat) wave equation

✷ =

∇α∇α.

However, this does not effect the statement of our main result which

is in fact quite simple:

Theorem 1.1 (Critical regularity for high dimensional Yang-Mills). Let the

number of spatial dimensions be even and such that n 6. Then for k0

n

2

and

0 ε0 suﬃciently small (depending on k0), there exists a fixed constant 0 C

such that the following holds: If (F (0),D(0),E(0)) is an admissible data set, with

Christoffel

symbols3

A(0) ∈

˙

H

n−2

2

∩

˙

H

k0+1,

and such that one has the smallness

condition:

(1.10) (F(0),E(0))

˙

H

n−4

2

A

ε0 ,

and if there exists constants Mk ∞,

n−4

2

k k0 such that:

(1.11) (F(0),E(0))

˙

H k

A

= Mk ,

then there exists a unique global solution to the field equations (1.3)–(1.4) with this

initial data and such that the following inductive norm bounds hold for all t:

F (t)

˙

H

n−4

2

A

Cε0 ,

F (t)

˙

H k

A

C(M

n−4

2

, . . . , Mk−1) Mk .

In particular the curvature F (t) remains smooth (in the gauge covariant sense) and

uniformly bounded for all times.

Remark 1.2. As alluded to we will prove the existence of a global (in space

and time) spatial Coulomb gauge in which the coeﬃcient functions of the curvature

F as well as the Christoffel symbols of the connection D are in the classical Sobolev

spaces

˙

H s. We will also show these quantities obey the expected range Strichartz

estimates.

1.2. Some Basic Notation

Here are some of the basic conventions used in this work, as well as an expla-

nation of the small constants which will be needed later. We use the usual notation

a b, to denote that a C · b for some (possibly large) constant C which may

change from line to line. Likewise we write a b to mean a C−1 · b for some

large constant C. In general, C will denote a large constant, but at times we will

also call C a connection. The difference should be clear from context.

In the body of our proof we will have use for a family of small constants which

satisfy the hierarchy:

(1.12) 0 γ δ 1 .

The constant denotes the size of the initial data in Theorem 1.1 with respect

to critical gauge covariant Sobolev norms. This is also the small constant used

throughout the paper when estimating norms involving the Yang-Mills connection

3See

the next Chapter for definitions.