The gauge fixing theorem with applications to the Yang-Mills flow over Riemannian manifolds

0.33 MB·

by  Min-Chun Hong

THE GAUGE FIXING THEOREM WITH APPLICATIONS TO
THE YANG-MILLS FLOW OVER RIEMANNIAN MANIFOLDS
7
1 MIN-CHUNHONG
0
2
Abstract. In 1982, Uhlenbeck [29] established the well-known gauge fixing
n
theorem, which has played a fundamental role for Yang-Millstheory. In this
a
paper, we apply the ideaof Uhlenbeck to establish aparabolic type of gauge
J
fixingtheoremsfortheYang-Millsflowandproveexistenceofaweaksolution

3 oftheYang-Millsflowonacompactn-dimensionalmanifoldwithinitialvalue
A0 in W1,n/2(M). When n = 4, we improve a key lemma of Uhlenbeck
] (Lemma 2.7 of [29]) to prove uniqueness of weak solutions of the Yang-Mills
G
flow onafourdimensionalmanifold.
D
.
h
at 1. Introduction
m
Let M be a compact n-dimensionalRiemannian manifold without bounday and
[
let E be a vector bundle over M with compactLie groupG. For a connection D ,
A

1 the Yang-Mills functional is defined by
v
1 YM(A;M)= |FA|2dv,
0 ZM
6 where F is the curvature of D . A connection D is called to be Yang-Mills if
A A A
0 it is a critical point of the Yang-Mills functional; i.e. D satisfies the Yang-Mills
A
0
equation
.
1
(1.1) D∗F =0.
0 A A
7 The Yang-Mills flow equation is
1
∂A
: (1.2) =−D∗F
v ∂t A A
i
X with initial condition A(0)=A , where A is a given connection on E.
0 0
r The Yang-Mills flow has played an important role in Yang-Mills theory. Atiyah
a
and Bott [1] introduced the Yang-Mills flow. Donaldson ([5],[6]) proved global ex-
istence of the smooth solution to the Yang-Mills heat flow in holomorphic vector
bundles over compact K¨ahler manifolds and used it to establish that a stable ir-
reducible holomorphic vector bundle E over a compact K¨ahler surface X admits
a unique Hermitian-Einstein connection, which was later called the Donaldson-
Uhlenbeck-Yautheorem,andsee differentapproachin [30]for the caseofholomor-
phic vector bundles over compact K¨ahler manifolds. Simpson [23] generalized the
Donaldson-Uhlenbeck-Yautheoreminholomorphicbundlesoversomenon-compact
K¨ahler manifolds. We refer to see [11], [18], [32] for further generalizations to the
Yang-Mills-Higgs flow on compact or complete K¨ahler manifolds.
When holomorphicvector bundles are notstable, there is a conjecture ofBando
and Siu [2] on the relation between the limiting bundle of the Yang-Mills flow and
Key words and phrases. Yang-Millsflow,Gaugefixingtheorem.
1
2 MIN-CHUNHONG
the Harder-Narashimhanfiltration on K¨ahler manifolds. The author and Tian [12]
established asymptotic behaviour of the Yang-Mills flow to prove the existence of
singularHermitian-Yang-MillsconnectionsonhigherdimensionalK¨ahlermanifolds.
Daskalopous and Wentworth [4] settled the Bando-Siu conjecture on K¨ahler sur-
faces. Recently, Jacob [16] and Sibley [24] settled the conjecture of Bando and Siu
[2] on higher dimensional K¨ahler manifolds by using the asymptotic result in [12].
Without the holomorphic structure of the bundle E overK¨ahler manifolds, it is
veryinterestingtoinvestigateexistenceoftheYang-Millsflowinvectorbundlesover
n-dimensional Riemanian manifolds. For the case of lower dimensional manifolds
(i.e. n = 2,3), Rado [19] proved global existence of the smooth solution of the
Yang-Millsflow. ItiswellknownthatYang-Millsequationsindimensionfourhave
many similarities to the harmonic map equation in dimension two, so dimension
fourisacriticalcaseforYang-Millequationsasdimensiontwoisforharmonicmaps.
Chang-Ding-Ye [3] constructed a counter-example that the harmonic map flow on
S2blowsupatfinitetime,soitwassuggestedthatYang-Millflowindimensionfour
shouldblowupinfinitetime. However,inacontrasttothesettingof[3],Schlatter,
Struwe and Tahvildar-Zadeh [21] proved global existence of the SO(4)-equivariant
Yang-MillsflowonR4. Later,theauthorandTian[13]alsoprovedglobalexistence
of the m-equivariant Yang-Mills flow on R4. Recently, Waldron [31] established
global existence of the smooth solution to the Yang-Mills flow when kF+k is
L2(M)
sufficiently small. When n > 5, it was known that the Yang-Mills flow could blow
up in finite time (e.g. [10]).
On the other hand, Uhlenbeck [29] established a gauge fixing theorem, which
has playedan important role to study the moduli space of Yang-Mills connections.
Since the Yang-Mills functional is gauge invariant, the Yang-Mills flow equation
(1.2) is not a parabolic system. In order to investigate existence of the Yang-Mills
flow,weapplytheideaofUhelenbeckin[29]toestablishaparabolicversionofgauge
fixing theorems, depending on time, such that the Yang-Mills flow is equivalent to
a parabolic system, which is called the the Yang-Mills equivalent flow (see below
(1.3-(1.4)). More precisely, we have
Theorem 1.1. For n≥4, let D beasmoothsolution of theYang-Mills flow(1.2)
A
in B¯ (x )×[0,t ] with smooth initial value A for some constant t > 0, where
r0 0 1 0 1
B (x ) is the ball in M with centre at x and radius r > 0. Assume that there
r0 0 0 0
exists a sufficiently small ε>0 such that
sup |F (x,t)|n/2dv ≤ε.
A
0≤t≤t1ZBr0(x0)
Then there are smooth gauge transformations S(t)=eu(t) and smooth connections
D =S∗(D ) satisfying the equation
a A
∂a
(1.3) =−D∗F +D s in B (x )×[0,t ],
∂t a a a r0 0 1
where
d
s(t)=S−1(t)◦ S(t).
dt
Moreover, for all t∈[0,t ], we have
1
(1.4) d∗a(t)=0 in B (x ), a(t)·ν =0 on ∂B (x ),
r0 0 r0 0
TheYang-Millsflow 3
1
(1.5) |a(t)|n/2+|∇a(t)|n/2dv ≤C |F |n/2dv
r2 a(t)
ZBr0(x0) 0 ZBr0(x0)
and
t1 1 ∂a t1
(1.6) |s|2+|D s|2+| |2dvdt≤C |∇ F |2dvdt.
r2 a ∂t a a
Z0 ZBr0(x0) 0 Z0 ZBr0(x0)
Wewouldliketopointoutthat(1.4)-(1.5)canbeobtainedbyusingUhlenbeck’s
gauge fixing theorem directly. However, since the Coulomb guage in Uhlenbeck’s
gauge fixing theorem might be not unique, one cannot prove (1.6) easily. Instead,
we have to follow all steps of Uhlenbeck’s original proof to fix Coulomb gauges for
each t>0 along the flow to prove (1.6).
As an application of Theorem 1.1, we prove
Theorem 1.2. For a connection A with F ∈ Ln/2(M) with n ≥ 4, there is
0 A0
a solution of the Yang-Mills flow (1.2) in M × [0,T ) with initial value D =
1 A0
D +A for a maximal existence time T >0. For each t ∈(0,T ), the solution
ref 0 1 1
A(t)isgauge-equivalenttoasmoothsolutionoftheYang-Millsflow. Atthemaximal
existencetimeT ,thereisatleastonesingularpointx ∈M,whichischaracterized
1 0
by the property that
limsup |F(x,t )|n/2dv ≥ε
i 0
ti→T1 ZBR(x0)
for any R∈(0,R ] for some R >0.
0 0
As a consequence of Theorem 1.2 for n = 4, it provides a new proof of local
existence of a weak solution of the Yang-Mills flow with initial value A ∈H1(M).
0
When n = 4, Struwe [26] proved existence of a weak solution, which is gauge-
equivalent to a smooth solution for t ∈ (0,T ) with the maximal existence time
1
T > 0, to the Yang-Mills flow in vector bundles over four manifolds for an initial
1
value A ∈ H1(M). The author, Tian and Yin [14] introduced the Yang-Mills α-
0
flowto provedthe globalexistence ofweaksolutions ofthe Yang-Mills flowonfour
manifolds. Recently, using an idea on the broken Hodge gauge of Uhlenbeck [28],
theauthorandSchabrun[15]establishedanenergyidentityfortheYang-Millsflow
at the finite or infinite singular time T .
1
It was known that Struwe [26] only proved uniqueness of weak solutions of the
Yang-Mills flow with initial value A ∈ H1(M) under an extra condition that A
0 0
is irreducible; i.e. for all s∈Ω0(adE)
ksk ≤CkD sk .
L2(M) A0 L2(M)
It has been an open problem about the uniqueness of weak solutions of the Yang-
Mills flow in four manifolds with initial data in H1(M) (Recently, this problem
was pointed out again in [31]). We would like to point out that the weak solution
constructedby Struwe in [26] is a weaklimit of smoothsolutions. In this sense, we
solve the problem of Struwe and prove
Theorem 1.3. When n = 4, the weak solutions of the Yang-Mills flow (1.2) with
initial value A ∈H1(M) are unique.
0
For the proof of Theorem 1.3, we need a variant of a parabolic gauge fixing
theorem for the Yang-Mills flow. However, in Theorem 1.1, d∗a = 0 in B (x )
r0 0
with Nuemann boundary condition a·ν = 0 on ∂B (x ) might be not unique, so
r0 0
4 MIN-CHUNHONG
the parabolic gaugefixing theorem in Theorem1.1 is not good enoughto establish
uniqueness of weak solutions of the Yang-Mills flow. To overcome the difficulty,
we improve a key lemma of Uhlenbeck (Lemma 2.7 of [29]) from the Neumann
boundary condition to the Dirichlet boundary condition. By a special covering
of M and ordering each open ball, we glue local connections together to a global
connection on the whole manifold M to prove uniqueness of weak solutions of the
Yang-Mills flow. Finally, we would like to remark that for n≥5, weak solutions of
the Yang-Mills flow with initial value A ∈H1(M) might not be unique (see [8]).
0
The paper is organised as follows. In Section 2, we recall some necessary back-
groundand estimates on the Yang-Mills flow. In Section 3, we prove Theorem1.1.
In Section 4, we prove Theorem 1.2. In Section 5, we show Theorem 1.3.
2. Some results on the Yang-Mills flow for smooth initial data
2.1. Local existence of the flow. Let D = D + A be a given smooth
A0 ref 0
connection in E, where D is a given smooth connection. We write D =
ref a(t)
D +a(t). Then
ref
F =F(D )+D a(t)+a(t)#a(t).
Da(t) ref ref
Following [26], we consider an equivalent flow
∂a(t)
(2.1) =−D∗ F −D (D∗ a)
∂t a(t) Da(t) a(t) a(t)
with a(0) = A . Note that (2.1) is a nonlinear parabolic system. By the well-
0
known theory of partial differential equations, there is a unique smooth solution
of (2.1) with the initial value on M ×[0,t ] for some t > 0. By the theory of
1 1
ordinary differential equations, there is a unique solution S ∈ G to the following
initial problem:
d
(2.2) S =−S◦(D∗a), in M ×[0,t ]
dt a 1
with initial value S(0)=I.
Through the gauge transformation
D =S∗D =S−1◦D ◦S,
a(t) A A
we have (e.g. see [26], [11])
F =S−1F S, D (D∗ a)=D ◦(D∗ a)−D∗ a◦D .
Da(t) A a(t) a(t) a(t) a(t) a(t) a(t)
Combining (2.1), (2.2) with above facts yields
d dS dD dS−1
D = ◦D ◦S−1+S◦ a(t) ◦S−1+S◦D ◦
dt A dt a(t) dt a(t) dt
= S−1 −D∗ F S
a(t) Da(t)
= −D∗(cid:16)F . (cid:17)
A A
This shows that D =(S−1)∗D satisfies the Yang-Mills flow with A(0)=A in
A a(t) 0
M ×[0,t ] for some t >0 and is unique (see [14]).
1 1
TheYang-Millsflow 5
2.2. Some estimates on the YM flow. We recall from [26] that
Lemma 2.1. Let A(t) be a smooth solution to the Yang-Mills flow in M ×[0,T]
with initial value A(0)=A for some T >0. For each t with 0<t≤T, we have
0
t ∂A 2
(2.3) |F |2dv+ dvds= |F |2dv.
A(t) ∂s A0
ZM Z0 ZM(cid:12) (cid:12) ZM
(cid:12) (cid:12)
Moreover,we have (cid:12) (cid:12)
(cid:12) (cid:12)
Lemma 2.2. Let A(t) be a smooth solution to the Yang-Mills flow in M ×[0,T]
with initial value A(0)=A , and assume that there is a constant ε>0 such that
0
sup max |F (·,t)|n/2dv ≤ε
A
0≤t≤Tx0∈MZBR0(x0)
for some positive R <1. Then there is a constant C such that
0
T T
(2.4) |∇ F |2dvdt≤C(1+ ) |F |2dv.
A A R2 A0
Z0 ZM 0 ZM
Proof. Applying the Bianchi identity D F = 0 and the well-known Weizenbo¨ck
A A
formula (e.g. [12]), we have
D D∗F =∇∗∇ F +F #F +Rm#F ,
A A A A A A A A A
where Rm denote the Riemannian curvature of M. Let {B (x )}J be an open
R0 i i=1
cover of M. By using the Ho¨lder inequality and the Sobolev inequality, we have
|∇ F |2dv ≤ |D∗F |2dv+C |F |3+|F |2dv
A A A A A A
ZM ZM ZM
2 (n−2)
J n n
≤ C |FA|n/2dv |FA|(n2−n2)dv
Xi=1 ZBR0(xi) ! ZBR0(xi) !
+C |D∗F |2+|F |2dv
A A A
ZM
1
≤ Cε2/n |∇ F |2dv+C (1+ )|F |2+|D∗F |2dv.
A A R2 A A A
ZM ZM 0
(2.4) follows from choosing ε sufficiently small and integrating in t. (cid:3)
Lemma 2.3. Let A(t) be a smooth solution to the Yang-Mills flow in M ×[0,T].
There exist constants ε=ε(E)>0 and R >0 such that if
0
sup max |F (·,t)|n/2dv ≤ε
A
0≤t≤Tx0∈MZBR0(x0)
for some positive R <1, then
0
t
(2.5) |FA|n/2(·,t)dv+ |FA|n−24|∇AFA|2dv
ZBR0(x0) Z0 ZBR0(x)
C t
≤ |F |n/2(0)dv+ |F |n/2(s)dv
A R2 A
ZB2R0(x) 0 Z0 ZB2R0(x0)
for all x ∈M.
0
6 MIN-CHUNHONG
Proof. Let φ∈C∞(B (x )) be a cutoff function with φ=1 in B (x ).
0 2R0 0 R0 0
Using the Yang-Mills flow equation and the Weizenbo¨ck formula, we have
∂F
A =−D D∗F =−∇∗∇ F +F #F +Rm#F ,
∂t A A A A A A A A A
where mboxRm is the Riemannian curvature. Then
d |FA|n/2φ2dv = n |FA|n−24 FA, ∂FA φ2dv
dt 2 ∂t
ZB2R0(x) ZB2R0(x) (cid:28) (cid:29)
=−n ∇A(|FA|n−24FAφ2), ∇AFA dv
2
ZB2R0(x)D E
+n |FA|n−24 hFA, FA#FA+Rm#FAiφ2dv
2
ZB2R0(x)
≤−n (|FA|n−24|∇AFA|2+ n−4|FA|n−24|∇|FA||2)φ2dv
2 2
ZB2R0(x)
+C |FA|n−24(|FA|3+|FA|2+ε|∇AFA|2)φ2+C|∇φ|2|FA|n/2dv.
ZB2R0(x)
Note that
|FA|n−24|FA|3φ2dv = |FA||FA|n2φ2dv
ZB2R0(x) ZB2R0(x)
2 n−2
n n
≤ |FA|n/2dv |FA|2(nn−22)φnn−2dv
ZB2R0(x) ! ZB2R0(x) !
≤Cεn2 |∇(|FA|n4φ)|2dv
ZB2R0(x)
≤Cεn2 |FA|n−24|∇|FA||2φdv+C |FA|n2|∇φ|2dv.
ZB2R0(x) ZB2R0(x)
Combiningaboveinequalitiesandchoosingεsufficientlysmall,theclaimisproved.
(cid:3)
Moreover,we have
Lemma 2.4. (ε-regularity estimates) Let A(t) be a smooth solution to the Yang-
Mills flow in B (x )×[t −r2,t ] with initial value A(0) = A for some r > 0
r0 0 0 0 0 0 0
and assume that there is ε>0 such that
sup |F (x,t)|n/2dv ≤ε.
A
t0−r02≤t≤t0ZBr0(x0)
Then there is some positive constant C such that
C t0
|F (x ,t )|2 ≤ |F |2dvdt.
A 0 0 r2 A
0 Zt0−r02ZBr0(x0)
Proof. By Proposition 3 of [12], we have
∂
( −∆ )|F |2+2|∇ F |2
M A A A
∂t
≤ C(|F |2+|F |)
A A
TheYang-Millsflow 7
Thenit isstandardtoapply the Scheon’sideato getthe requiredresult(e.g. [12]),
so we omit the details here. (cid:3)
Lemma2.5. (Higher regularityestimates) LetA(t) beasolution totheYang-Mills
flow in M ×[0,T] with initial value A(0) = A . Assume that there is a constant
0
ε>0 such that
sup |F (x,t)|n/2dv ≤ε
A
0≤t≤TZBr0(x0)
for a ball B (x ), Then there are positive constants C(k) for any integer k ≥ 1
r0 0
such that
T
|F |2+···+|∇kF |2dvdt≤C(k).
A A A
Zr02 ZB12r0(x0)
Proof. By Proposition 3 of [12], we have
∂
( −∆ )|∇kF |2+2|∇k+1F |2
∂t M A A A A
k
≤ C|∇kF | (|∇k−jF |+1)≤C(|∇kF |2+1).
A A A A
j=0
X
Then we apply the Moser’s estimate to get the required result. (cid:3)
3. Proof of Theorem 1.1
In order to prove Theorems 1.1, we need some lemmas.
Lemma 3.1. For a matrix u1(t), set s1(t)=e−u1(t)◦deud1t(t). If |u1(t)| is bounded,
then
du du
1 1
(3.1) |∇s (t)|≤C ∇ +C|∇u | .
1 1
dt dt
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Proof. Note that (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
deu1(t) deu1(t)
∇s (t)=∇e−u1(t)◦ +e−u1(t)◦∇
1
dt dt
and
u2 uk
eu1(t) =I +u (t)+ 1 +···+ 1 +··· .
1
2! k!
Then
deu1(t) = du1 + ddut1u1(t)+u1(t)ddut1 +···+ ddut1u1k−1+···+u1k−1ddut1 +··· ,
dt dt 2! k!
which implies
deu1(t) ≤ du1 + du1 (e|u1(t)|−1).
dt dt dt
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Similarly, we have (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
∇e−u1(t) ≤|∇u |+|∇u |(e|u1(t)|−1).
1 1
(cid:12) (cid:12)
Then (cid:12) (cid:12)
(cid:12) (cid:12)
8 MIN-CHUNHONG
deu1(t)
∇
dt
du ∇du1u (t)+u (t)∇du1 ∇du1uk−1+···+uk−1∇du1
= ∇ 1 + dt 1 1 dt +···+ dt 1 1 dt +···
dt 2! k!
du1∇u (t)+∇u (t)du1 du1∇uk−1+···+∇uk−1du1
+ dt 1 1 dt +···+ dt 1 1 dt +··· .
2! k!
Moreover
|∇(uk)|=|∇u uk−2+···+uk−2∇u |≤(k−1)|∇u ||u |k−2.
1 1 1 1 1 1
Then
deu1(t)
∇
dt
(cid:12) (cid:12)
≤ (cid:12)(cid:12)(cid:12)∇du1 +(cid:12)(cid:12)(cid:12)∇du1 (e|u1|−1)+|du1||∇u1|(1+|u1|+···+ |u1|(k−2) +···)
dt dt dt (k−2)!
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) du(cid:12) (cid:12) du(cid:12) du
≤(cid:12)(cid:12) ∇ (cid:12)(cid:12)1 (cid:12)(cid:12)+ ∇ (cid:12)(cid:12)1 (e|u1|−1)+| 1||∇u1|e|u1(t)|.
dt dt dt
(cid:12) (cid:12) (cid:12) (cid:12)
This prov(cid:12)(cid:12)es our(cid:12)(cid:12)clai(cid:12)(cid:12)m. (cid:12)(cid:12) (cid:3)
(cid:12) (cid:12) (cid:12) (cid:12)
Lemma 3.2. For a given function f ∈ W1,p(Sn−1×[0,1]), let v be a solution of
the heat equation on Sn−1×[0,1] satisfying
∂rv =∆Sn−1v+f
with v(θ,1) = 0 on Sn−1. Let ϕ(r) be a smooth cut-off function in [0,1] with
ϕ(r) = 1 near 1 and ϕ(r) = 0 for [0,η] with some small constant η > 0. Then we
have
(3.2) kϕvkW1,p(Sn−1×[0,1]) ≤CkfkLp(Sn−1×[0,1])
and
(3.3) kϕvkW2,p(Sn−1×[0,1]) ≤CkfkW1,p(Sn−1×[0,1])
for any p>1.
Proof. ThislemmawasmentionedbyUhlenbeckin[28]. Forcompleteness,wegive
a proof here.
(3.4) ∂r(ϕv)=∆Sn−1(ϕv)+ϕf +∂rϕv.
By the standard Lp-estimate of parabolic equations [17], we have
|∂ (ϕv)|p+|∇2(ϕv)|p+|∇ (ϕv)|pdθdr ≤C |ϕf +∂ ϕv|pdθdr
r θ θ r
ZSn−1×[0,1] ZSn−1×[0,1]
≤C |f|pdθdr+C |v|pdθdr.
ZSn−1×[0,1] ZSn−1×[η,1]
Then we claim that there is a constant C such that
|v|pdθdr ≤C |f|pdθdr.
ZSn−1×[η,1] ZSn−1×[0,1]
If not, there is a sequence of k and solutions v of the heat equation
k
∂rvk =∆Sn−1vk+fk
TheYang-Millsflow 9
with v (θ,1)=0 on Sn−1 such that
k
|v |pdθdr ≥k |f |pdθdr.
k k
ZSn−1×[η,1] ZSn−1×[0,1]
Set
v f
v˜ = k , f˜ = k .
k k
kvnkLp(Sn−1×[η,1]) kvkkLp(Sn−1×[η,1])
It implies that kv˜kkLp(Sn−1×[η,1]) =1 and v˜k satisfies
∂rv˜k =∆Sn−1v˜k+f˜k.
withv˜k(θ,1)=0onSn−1. Notingthatkf˜kkLp(Sn−1×[η,1]) ≤1/k,thereisafunction
v such that as k → ∞, ∂ v˜ converges to ∂ v˜ and ∇2v˜ converges to ∇2v
∞ r k r ∞ θ k θ ∞
weakly in Lp(Sn−1 × [η,1]). By the Sobolev compact imbedding theorem (e.g.
[17]), v˜ converges to v strongly in Lp(Sn−1×[η,1]), which implies that
k ∞
kv∞kLp(Sn−1×[η,1]) =1.
Moreover, v also satisfies
∞
∂rv˜∞ =∆Sn−1v˜∞.
with v˜ (θ,1) = 0 on Sn−1. By the backward uniqueness of the heat equa-
∞
tion, v˜ must be zero in Sn−1 ×[η,1]. This is contradicted with the fact that
∞
kv∞kLp(Sn−1×[η,1]) = 1. Therefore, our claim is proved, so (3.2) holds. Similarly,
(3.3) can be proved by differentiating in r in (3.4).
(cid:3)
We recall a key lemma of Uhlenbeck (Lemma 2.7 in [29]) in the following:
Lemma 3.3. For somep> n, let A∈W1,p(U) bea connection satisfying d∗A=0
2
in U¯ =B¯ (0) with
1
kAk ≤k(n)
Ln(U)
for a sufficiently small k(n). Let λ∈W1,p(U) satisfy λ·ν = 0 on ∂U. There is a
small constant ε>0 such that if
kλk ≤ε,
W1,p(U)
then there is a gauge transformation S =eu ∈W2,p(U) to solve
(3.5) d∗a=d∗(S−1dS+S−1(A+λ)S)=0
in U with udx=0 and ∂ u=0 on ∂U.
U ν
Now weRcomplete a proof of Theorem 1.1
Proof. Without loss of generality, we assume that U = B (0) and denote D =
1 A
d+A. At t=0, itfollows fromUhlenbeck’s gauge fixing theorem[29]that there is
a smooth gauge transformation S = S(0) and a connection D = S∗(D ) =
0 a(0) 0 A(0)
d+a(0) satisfying
d∗a(0)=0 in U, a(0)·ν =0 on ∂U
and
|a(0)|p+|∇a(0)|pdv ≤C(p) |F |pdv
a(0)
ZU ZU
for any p≥ n.
2
10 MIN-CHUNHONG
For any p∈(n/2,n]and forthe aboveε>0,there is a constantδ >0 suchthat
for all t,t′ ∈[0,t ] with |t−t′|≤δ, we have
1
(3.6) |∇(A(t)−A(t′))|p+|A(t)−A(t′)|pdv ≤εp.
ZU¯
Next, we follow the procedure of [29] to fix a Coulomb gauge in [0,δ]. Through
the gauge transformation S the induced connection D = S∗(0)(D ) = d+
0 A˜(t) A(t)
A˜(t), with A˜(t) = S−1dS +S−1A(t)S , is also a smooth solution the Yang-Mills
0 0 0 0
flow in U¯ ×[0,t ] with A˜(0) = a(0). However, A˜(t) does not satisfy the boundary
1
conditionofA˜·ν =0on∂U,sowecannotapplyaboveLemma3.3tofixaCoulomb
gaugeforA˜(t) fort∈[0,δ]. Inorderto sortoutthe boundaryissue,itfollowsfrom
Lemma 2.6 of [29] to get that there are gauge transformations eu1(t) such that
(e−u1(t))∗(DA˜(t))=e−u1(t)◦(d+A˜(t))◦eu1(t) =d+a1(t),
where a (t):=A˜(0)+λ(t) and
1
(3.7) λ(t)=−A˜(0)+e−u1(t)deu1(t)+e−u1(t)(A˜(t))eu1(t).
In fact, we can choose u (t)=ϕv˜, where ϕ(r) is a smooth cut-off function defined
1
in Lemma 3.2 and v˜ is the solution of
∂
(3.8) (∂r −∆Sn−1)v˜=x·(A˜(t)−A˜(0)) for (r,θ)∈[0,1]×Sn−1
with v˜(1,θ)=0 for all θ ∈Sn−1. Then, we have u1(t)=0 and deu1(t) =du1(t) on
∂U for all t∈[0,δ], which imply
λ(t)·ν =(du (t)+A˜(t)−A˜(0))·ν =0 on ∂U,
1
which implies that the new connection a (t) satisfies the requiredboundary condi-
1
tion a (t)·ν =0 on ∂U.
1
Noting that A˜(t)=S−1dS +S−1A(t)S , we have
0 0 0 0
A˜(t)−A˜(0)=S−1(A(t)−A(0))S .
0 0
Using (3.6), we have
(3.9) kA˜(t)−A˜(0)k =kA(t)−A(0)k ≤ε
W1,p(U) W1,p(U)
for any t∈[0,δ].
By the Lp-estimate in Lemma 3.2, we have
(3.10) |∇u |q(t)dv ≤CkA˜(t)−A˜(0)k ≤CkA˜(t)−A˜(0)k
1 Lq(U) W1,p(U)
ZU
forq >n. BytheSobolevimbeddingtheorem,|u (t)| isuniformlyboundedforany
1
t∈[0,δ] for a sufficiently small δ >0. Moreover,differentiating equation (3.8) in t
yields
∂u (t) ∂ ∂A˜
∂1t =ϕ(∂r −∆Sn−1)−1(x· ∂t ).
By applying the Lp-estimate in Lemma 3.2 again, we have
∂u (t) ∂A
|∇ 1 |2dv ≤C | |2(·,t)dv ≤C |∇ F |2(·,t)dv
A A
∂t ∂t
ZU ZU ZU
for any t∈[0,δ].