manuscripta math. 43, 131 - 166 (1983) manusc ripta mathemati ca �9 Springer-Verlag 1983

A MONOTONICITY FORMULA FOR YANG-MILLS FIELDS

Peter Price

"Monotonicity formulae" have been useful in

the proof of regularity theorems in minimal surface

theory (see e.g. [3, i] and for energy-minimizing

harmonic maps [ii]. Here we derive an analogous formula

(Theorems i, i') for (stationary) Yang-Mills fields. A

Liouville type vanishing theorem (Corollary 2) follows

immediately. In a preliminary report on this work [9]

we included the derivation of a similar result for

harmonic maps (called "nonlinear sigma models" in the

physics literature) but we have been informed that this

is an earlier result due essentially to Gather,

Ruisjenaars, Seiler and Burns [5] (see also [6] Theorem

8.12 and the following remarks, and [i0]). For

completeness we include here the statements of the

results for harmonic maps.

Following Allard [I] we can derive similar

formulae (see Theorems i", l"a for the precise

statements) requiring not that the Yang-Mills action (or

131

PRICE

energy of the map) be stationary, but only that the

generalised quark current density (or generalised

tension vector field), as defined below, be suitably

bounded.

Defining densities

~im 4-n i ( )12 O(x) : r+0 r fB (x) where ~(m) is the

r

curvature of the connection ~ (or

~im 2-n 12 8(x) : ri0 r fB (x) Idu

r

where

in the harmonic map case,

u is the map) it follows (Corollary 3) that with

suitable assumptions, the n - 4 (or n - 2)

dimensional Hausdorf measure of the positive density set

vanishes. Here n is the dimension of the base

(domain) manifold M . One might ask whether the

positive density set is the singular set, in a suitable

sense. We are a long way from answering such a question.

We would like to thank Leon Simon for useful

discussions and S. Hi!debrandt for bringing [5], [6]

and [i0] to our attention.

NOTATION. We recall some facts about the Laplacian

acting on section-valued forms [2]. If V is a

Riemannian vector bundle over the Riemannian manifold M ,

i.e., V has a covariant derivative (connection) and a

fibre metric such that parallel transport is an isometry,

we write V V for the connection on the bundle V and

V M for the Riemannian connection on TM . Then these

connections induce the tensor product connection

V = V M ~ V v on all associated tensor product bundles.

132

PRICE

An example is the associated bundle AkT*M Q V whose

sections, called section-valued k-forms, we denote

ek(M, V) . We derive various other differential

operators on ek(M, V) as follows. Note that

V : ek(M, V) § el(M) ~ ek(M, V) .

The L 2 adjoint of V , denoted V* , is given by

V* : el(M) ~ sk(M, V) § ek(M, V)

V* = - trace o V

The exterior covariant derivative

d V : ek(M, V) § ek+l(M, V)

d V = A o V

is the projection by exterior product of V .

its L 2 adjoint

6 V k-i(M : ek(M, V) § e , V)

6 V= -~o~oV

We denote

where

: el(M) ~ sk(M, V) § el(M) ~ s V)

is the metric isomorphism and ~ is the projection by

interior product. Note that

133

PRICE

d V = V on c~ V) = F(V)

and

6 V = V* on gI(M, V) = s F(V)

If V is a trivial bundle with flat connection V V

then d V and 6V reduce to d and 6 respectively.

The laplacian A V is defined

A V : dV6 v + 6Vd V

A V : sk(M, V) § ~k(M, V)

A section-valued form ~ ( sk(M, V) satisfying

AV~ : 0

is called harmonic. If M is compact without boundary

this is equivalent to

dV~ : ~V~ : 0 .

Let P(M, G) be a principal bundle with

compact structure group G over a Riemannian manifold

M . Ad P is the adjoint bundle

Ad P = P XAdG=g =

with g the Lie algebra of G .

of smooth connections ~ on P

134

Let U be the space

Every such connection

PRICE

induces a covariant derivative V P on Ad P We also

have the Riemannian connection V M on the tangent bundle

TM , and the tensor product connection on ek(M, Ad P) .

All these connections we denote V An Ad G invariant

inner product on g induces a fibre metric on Ad P

making Ad P and all tensor product bundles such as

AkT*M ~ Ad P into Riemannian vector bundles over M

The space of connections U is an affine

space. In fact

(U) ~ SI(M, Ad P)

This structure allows us to define Sobolev spaces of

connections. If W k'p is the Sobolev space of functions

with k derivatives in L p we can define wk'P(v) ,

the Sobolev space of sections of a Riemannian vector

bundle V . Similarly

Also wk'P(v) is, as usual, the closure in wk'P(v) o

the compactly supported C sections of V These

spaces are independent of the connection ~ on V .

denote W k 'p (AqT*M~Ad P]

the space of W k'p

U k'p , is

of

We

by wk'p(gq(M, Ad P)] Then

connections on P(M, G) , denoted

135

PRICE

where ~ ( U is fixed. The affine structure of U

makes this definition independent of the choice of base

connection Similarly we define U~oP ' :o ~'p etc.

The curvature ~ 6 e2(M, Ad P) of a connection m ~

is given by

= ~(~) = d V2 = d V o V

Here Ad P acts on itself by the fibrewise Lie bracket

inherited from ~ . If ~' = ~ + B ( U2 '2 A U9 '4

~(~') = ~(~) + dVB + [B, B]

and Ad P))

The bundle of groups ad P is defined

ad P = P XadGG .

The C ~ @au@e @roup ~ = F(ad P) acts on the

connections by conjugation. If s (

-i s-iVs s*(V) : s o V o s : V +

Here s-iVs 6 el(M, Ad P) is defined as follows. Let

~s be the component function of the section s 6 F(ad P)

~s : P(M, G) + G

136

PRICE

such that ~s(Ua) : ad(a-1)~s(U) for all u ( P ,

a ( G . If we write u~ for the element of ad P

given by the equivalence class of (u, ~) ( P x G

s(x) : U[s(U)

then

for all u (~-l(x) , where ~ is the projection of

the bundle P . Let X* be the horizontal lift of a

vector field X on M , and @ the canonical 1-form

on G. Then

8 o dSs(X* ) : p §

and

@ o d~s(X*)(ua ) : Ad(a-l]@o d~s(X*)(u).

Thus 8 o d~s(X*) defines a section of

-i s Vs ( el(M, Ad P) is given by

s-iVs(X) = u[@odSs(X*)(u)] for all

Ad P . Then

X (el(M) .

If ~0' : ~ + B , V' : V + B

s-iVs -i s*(V') = V + + s o B o s

The appropriate group of gauge

transformations for the U 1'2 N U_ ~ connections are

the W 2'2 n W 1'4 sections of ad P These we denote

~2,2 A_G 1'4 . Note that for dim M _ < 3 the Sobolev

137

PRICE

embedding theorem guarantees these gauge transformations

to be continuous and thus preserve the topological

structure of the bundle. This is not necessarily so

for dim M ~ 4 , which is the case of interest here.

For ~ ( U_ 1'2 N U~ '4 and s ( G2,2 N GI,4 ,

s*(~) ( U 1'2 A U ~ and ~(~) ,

~(s*(~)) ( W~ Ad P)) with

-I ~(s*~) : s o r i o s

The Yang-Mills action functional, for

( U 1 '2 r~ Uf '4 , i s given by

s(co) = I1~(co)1122 = fM (~(co) , ~(co))dV = fMLaO)

If ~ = ~ + B for some smooth base connection o o

the Lagrangian density i(~) is given by

L(~) = ln (%) + dVB + [B, B]I 2

Also we define

C(~) = {~' = ~+B(ul'2nU_ ~ : BEWI '2NW~ Ad P) ]} . o o

Then a Yang-Mills connection is a critical point of

SIC(~ ) , i . e . , co ( U 1 '2 N_U_ ~ is Yang-Mills if

< = 0 dt t=O

138

PRICE

for all 1-parameter families t (C(~) such that

o : ~ and S(t) is differentiable in t . The

corresponding curvature ~(~) is called a Yang-Mills

field.

A C 2 Yang-Mills connection ~ satisfies

the Euler-Lagrange equations

~V~ = 0

For any C 2 connection we also have the Bianchi

identity

dV~ = 0

Thus

E C2(~) is Yang-Mills = s : 0

i.e., ~ is a harmonic section-valued 2-form in

~2(M, Ad P)

We recall the analogous definitions for

harmonic maps [ii]. Let M and N be smooth

Riemannian manifolds of dimension n and k'

respectively, and suppose N is isometrically embedded

in ~k for some sufficiently large k Let

WI,21M, ~k) be the separable Hilbert space of maps

u : M § ~k whose component functions have first

W1,2 (M, ~k) derivatives in L 2 Similarly we define ~oc

139

PRICE

and W I'2(M IR k ) 0

W~oc(M ,1,2. N) and WI'2(M,o N)

the appropriate W 1,2 (M, IR k]

a.e. x(M

Then the spaces WI'2(M, N) ,

consist of those maps u in

space such that u(x) E N

given by

For u ( WI'P(M, N) the energy functional is

2 = f (du(x), du(x))dV = fM E<u) = Hdull 2 M e(u)

In terms of a local orthonormal frame field for M the

Lagrangian density e(u) is given by

e(u) = (du(ei], du(ei] ] (summation convention)

where the inner product is that of Tu(x)N (or of IRk).

Note that du is a section-valued one-form on

du ~ sI(M, u*TN) , the one-forms on M

sections of the pull-back bundle u*TN

Given u E WI'2(M, N) we define

M

with values as

over M .

C(u) = {r 6 WI'2(M, N) : u - r 6 WI'2(M, IRk]} O

Then a hoarmonic map is a critical point of

i . e . , u ( WI '2(M, N) i s h a r m o n i c i f O

ElC(u ) ,

d E( t) l = o dt t=o

140

PRICE

for all 1-parameter families ~t (C(u) such that

G ~ = u ~ and [(~ t] is differentiable in t .

A C 2 harmonic map satisfies the Euler-

Lagrange equations

~Vdu = 0

This a quasi-linear second-order elliptic system for the

map u . For any u ( C2(M, N) we also have the

equation ("Bianchi identity")

dVdu = 0

due to the vanishing of the torsion of N .

dVdu = u*T , with T the torsion tensor of

(In general

N .) Thus

u ( C2(M, N) is harmonic = AVdu : 0

i.e., du is a harmonic section-valued 1-form in

el(M, u*TN)

MONOTONICITY FORMULA. We now define C~ c C(~) the

variations of ~ by (1-parameter) reparametrisation of

M If ~ is a compactly supported C I diffeomorphism

of M and we write ~ : ~ + B then ~*B does not o

belong to el(M, Ad P) : for X ( T M x x

,B(Xx] : )

141

PRICE

(with z the projection of Ad P) We need to

construct a l~ts ~* of to el(M, Ad P)

In the special case when { }t((-g,s) is a 1-parameter

family with ~o : the identity we can construct a

lifting by parallel transport along x t = ~t(x)

do this by lifting ~t to the principal bundle

defining

We

P(M, G) ,

~t(u) = T~(u) for all u E ~-l(x)

o is the parallel transport, with respect to an where T t

arbitrary smooth connection, along the curve x s = ~S(x)

from x ~ = x to x t = ~t(x) . Then we define

~t = ~*~

where the connection ~ is regarded as a g-valued

1-form on P . Then t is a connection (of the same

class as ~), as parallel transport commutes with right

t multiplication in the bundle. The curvature of ~ ,

which we denote ~t when we regard it as a horizontal~

equivariant g-valued 2-form on P , is, by the structure

equation

142

PRICE

~J = da t + [a t, a t ]

: qbt'={da + [a, a]}

Then as a section-valued 2-form on M ,

~t ( r Ad P) , and

f~t(Xx, yx ] = u~t(Xu, y

,~., = <' YO u~ [~,Xu ' t-'-

= Tot ((~t(u))

[< = T t t(u ) o

Tt~ r t ty : o lr215 r x)

= (Tto ~t*Q 1 (Xx, Yx ]

t * , ty*~ ~(r ** uJ

f. tx, t --

X* Y* Here , are any lifts to T P for u (~-l(x) u u u

of the v e c t o r s Xx' Yx ( TxM . The f i r s t e q u a l i t y i s by

the definition of associated vector bundles, the fourth

i s by t he d e f i n i t i o n of p a r a l l e l t r a n s p o r t i n a s s o c i a t e d

vector bundles and the fifth uses that *~X~,. is a lift

of ~atx,, x to T ~ ( u ) P Thus

143

PRICE

a t : m t O

%t*~ s2( o ( M, Ad P) .

If W : W + B we define ~t(B) by O

t t* 0

Then ~t provides the required lift of ~t to

el(M, Ad P) Thus we define C~ , the variations

of w by 1-parameter reparametrisation of M (called

r-vars of w) as

: wt ~t C~ {w t 6 C(w) : = ~ w for a 1-parameter

family of compactly supported C I

diffeomorphisms of M with ~o : the identity}

Note that C~ depends on the smooth connection w O

chosen to d e f i n e the l i f t ~t o f ~t

We calculate the first variation of the Yang-

Mills action for variations of w ~ U 1'2 A U ~ by such

~t

s(t] : ]M (W, t]dV

t "~" "%

M . . . . = , dV

144

PRICE

as parallel transport with respect to any connection is

an isometry. Using a change of coordinates this becomes

s(~ t)

= <t x) x)

t-i t-I

~t(M):M[ x[

x jilt l]~x~dV~x~

as

t-i

where J[~t-l)(] is the Jacobian of ~ t-1 By ~ ... X

we mean the same expression as in the first factor of the

inner product. Here {el} is a local i=l, . . . ,n

orthonormal frame field for TM , I~l 2 :

(~(ei, ej], ~(ei, ej]) , we are using the summation

convention and the sums over i and j are for

1 _< i < j _< n Differentiating, we have

ds(Lot] t=o =-fM {l[~I2div X+4(~([X'ei]'ej]'~(ei'ej])}dV

where X(x)= d~[~t(x)) t=O and div X = (Ve.X , el] l

145

PRICE

Since

(IX, eli , ek)= [Vxei, ek)- (Ve X , ek] i

= - (el, Vxe k] - (Ve.X, e k] i

due to the orthonormality of the {eli , we obtain the

first variation formula

(1) d~S(~ t] t=O=-fM{l~12div X-4(~(VeiX,ej],~(ei,ej]]]dV

In the harmonic map case we can define

C~ a C(u) as

t }t }t C~ = {u t (C(u) : u = u o for a 1-parameter

family of compactly supported C I

with ~o : the identity.]

diffeomorphisms of M

For such u du t ~t"du , and

.T.

E(ut] = fM ( ~ t*du' ~ t''du] dV.

Changing coordinates as in the Yang-Mills case we obtain

the analogous first variation formula

(2) d-~t (u t] It=O = -fM{Idul2div X-9(dU(Ve.X],du[ei]]}dV . 1

146

PRICE

REMARK. In the harmonic map setting the map u : M § N

defines the bundle u*TN and thus d(u o ~) is

automatically a section-valued 1-form - we don't need to

construct a lifting. In the Yang-Mills setting the

bundle is fixed.

Initially we restrict to the case where M

has constant sectional curvature - b 2 Ibm_0) Let

Bo(x) be the geodesic ball of centre x and radius O

in M , and let

for ~ ( U 1'2 A U ~

the ball B (x) , and

be the Yang-Mills action of ~ in

= fB . (x)Idul2 a

for u ( WI'2(M, N) , be the energy of u in this ball.

We assume the distance from a point p ( M to the cut

locus or boundary of M is at least i Also

dim M = n

THEOREM 1. Let ~ ~ ~1,2 n ~0'4(Bl(p) J ~ ~

connection, and n >__ 4 . Then

be a Yang-Mi l ls

(3) 04-ns~(~) -< p4-ns~(~)

for x ~ B�89 0 < (7 _< p _< �89 .

147

PRICE

REMARK. The analogous statement (see [6]) for a

harmonic map u (WI'2(BI(P), N] , for n { 2 and the

same restrictions on x , ~ and P , is that

(• 2 - n ~ - - x ~ (4) o2-nE (u) _< P E <u).

The following corollary is immediate:

U !'2 n U ~ be a Yang-Mills COROLLARY 2. Let W ~ =~oe =~oc

connection with M = ~n or H n (hyperbolic space of

constant negative sectional curvature) and n ~ 4 . If

SR(~O) = oIR n-4] as R + ~ for some x ~ M

then ~ is the flat connection.

REMARK. The corresponding Liouville theorem in the

harmonic map setting is that ER(U) : o(R n-2] implies

that u is constant.

NOTE. (i) In Corollary 2 we only need that

UI, 2 ~ U~ 4 gauge equivalent to a connection in :~oc =~oc

is

(ii) The following proof of Theorem I shows that

we only need w to be stationary under repara-

metrisations of M (r-stationary).

PROOF. Let {~t~t((_E,~ ) be a 1-parameter family of

compactly supported cl-diffeomorphisms of BI(p) such

148

PRICE

that

d t:0 D dt ~t(y) = ~(r)r D-~

where r is the geodesic radial coordinate on B�89

~(r) will be chosen to approximate the characteristic

function of the interval [o, c] for o < �89 Let

"or ~-~' ei]i=l,. ~ ,n-i be an orthonormal basis for T M B

�9 . y

Then

V D x : Dr Dr

and

D D V x : [rV e. e. Dr [Vre. ~-r " 1 i i

Now

D V --- = (br coth br)e. re. Dr i

i

when the sectional curvature of M is - b 2 , and we

take br coth br = i when b : 0

Using this choice of X in the first

variation formula i,

149

PRICE

0 = ~t s(~t] t:O : - fM [ l~12{(e~)' + ~(n-l) br coth br}

- 4(~r)' ( ~ ( ~ , ej), ~(~-~-~, ej))

- 4~ br coth br(~(ei, ej), ~(ei, ej]]]dV

Thus

- fM ~ ' r l a l 2 + (4-n) fM ~1~12 = - 4 fM r~' I ~ J al 2

+ fM ~{(n-5)(br coth br-Z){e[ 2

+ 4(br coth br-l)I~ ~ ~I 2}

We choose, for T ( [~, p] , ~(r) : ~y(r) : r with

r smooth and satisfying r : i for r ( [0, i] ,

r = 0 for r ( [l+s, ~) , e > 0 and r ~ 0

Then

Thus

2} = ~ {IM ~lr~l --,- (4-n) fM ~;'~ Ir~12 4~{./'M '~TI@ 2 ~12].

+ f l I ~T{ (n -5 ) (b r coth br-1) I r~ l 2

+

150

PRI CE

99T {T4-n fM q 1~12} = 4T4-na~a {r.M q i~a j ~! 2}

3-n { + Y fM ~T (n-5)(br coth br -1) la l 2

+ 4(br coth br-1)1~-~ 1 el2}. The theorem is, of course, trivial for n -- 4 For

n >_ 5 the right hand side of the above equality is

positive. (3) follows by integrating over [d, p] and

taking the limit ~ § 0

REMARK. In fact we obtain the equality

( 5 ) p4-n fB ( x ) 1~12 P

4-n [2

4-nl~ 2 = 4 fBo(x~B~(x ) r J ~I

+ f~ d, a - n fB (x) 1:

(n-5)(br coth b r - l ) I~ I 2

REMARK. Corresponding to (5) we have, in the harmonic

map case

151

PRICE

(6 ) p2-=/L ( tdul 2 2-n 2 x) - ~ fB (x) ldul g

= 2 /Bp(X)~B (x) r2-nj~U~r 2

+ /a p dT T l-n /BT(x) <(n-3)(br coth br-!)JduJ 2

~u12 } + 2(br coth br-l)I-~r .

REMARK. If b 2 = 0 then equality holds in 3 (4) if

and only if

~ r

in the annulus Bp(X) ~ B~(x) .

If - b 2 < 0 and n = 5 in 4

in 3) then equality holds if and only if

( o r n = 3

~r-

in Bp(X)

Note that if r denotes the multiplication

map in normal coordinates on Bl(O) for r ! ! ,

r : x +rx

and if ~ is a connection on a bundle P over the unit

152

PRICE

-i sphere DBI(O) then r *m is a connection on the pull-

back bundle r-l*P over Bl(O) - {o} which satisfies

J = o Dr

sn-i Any smooth non-trivial Yang-Mills field on (n ~ 5)

~n gives rise to a Yang-Mills field on with an isolated

singularity at the origin. For such a field

S;(m) = KRn-4/(n-4)

with K = SI0~ sn_ll, the action of the original % J

sn-i connection on The analogous statement holds for

S n smooth non-constant harmonic maps u : § N (n >- 2)

We now relax the condition that M be of

constant non-positive curvature. Then in the notation

used in the above proof we have, with x = (r, @) ,

(Vre D r e (r' @)dr' �9 Dr' ej)(x) = ~ij + fo ij ' 1

with

D D Eij(X) : ~c~ (Vre. --Dr' ej)(x) .

1

We assume

l~ij(x)l ~ A for x E B!(p)

153

PRICE

Then

div X Z ~'r + n~ - (n-l)~Ar

and

4(~(Ve.X , ej], ~(ei, ej)] i

_< 4~'rl~ r d r~l 2 + 4~lal 2 + 4(n-l)~rA]~l 2

As in the above proof we obtain

eC(n)ATT4-n ~T $ {~M ~ [~12} + (4-n)eC(n)ATT3-n I'M ~T [~1 2

+ (l+~)c(n)AeC(n)ATT4-n ~M ~ 1~12

_ 4eC(n)AT 4-n ~ I ~-~ d ~12} > T ~{IM ~T Dr >_ 0

Thus

cAo 4-n f 2 (7) e 0 I~I B(x)

cAr 4-n, + fBo(x)~Bo(x) 4e r -~ ] al 2

-< ecApp4-n SB (x) 1 12 P

for x (B�89 and 0 < ~ _< 0 -< �89 �9 This establishes

THEOREM i' We assume the conditions of Theorem I, but

154

PRICE

allowing the manifold M to be arbitrary. Then (7)

holds, and in particular

cAp 4-n~x, . (S') 04-ns (68) ~ e p ~pk68)

with c , A , ~ , p and x as given above.

REMARK. By a change of scale (see e.g., [ii]) A can

be made arbitrarily small.

A similar monotonicity formula holds, as we

now show, without requiring that the Yang-Mills action

of 68 be stationary (or even r-stationary) but merely

that the generalised source density which we define

below, be suitably bounded.

Firstly we note that if, as above, 68t is a

1-parameter variation of a connection 68 given by

t ~* 60 : ~ 68

with the lift ~ of ~t defined by parallel transport

with respect to the smooth connection 68 , with o

68 : 68 + B for some B 6 W 1'2 n W~ Ad P)) , then o

[d68t/dt]t=0, , denoted ~ , (.W~ Ad P))

is given by

(8) ~ : L ~ = X ~ ~ + VB(X)

X

155

PRICE

with X (X) the initial velocity vector field of

~t (~] Here L is the Lie derivative and V

covariant derivative in e~ Ad P) defined by

Recall that

is the

.

o , .] Vy = Vy + [B(Y)

0 on s (M, Ad P) , with V ~ the covariant derivative

defined by ~ For B ( W 2'2 n WI'4(~I(M, Ad P)) the O

term VB(X) is an infinitesimal gauge transformation.

In general, if t 6 C(w) is a 1-parameter

variation of ~ : o with ~ ( W 1,2 N W~ Ad P))

�89 S(~ot) t=0 = (~(~)' d ~>

: SM dye] dv

we use < , > to denote the L 2 inner-product on

ek(M, Ad P)]. We define the first variation T of the

Yang-Mills action of a connection ~ , 6S , as a

linear functional on CI(M, Ad P) given by

for all

6S (B) : 2<~(~), dVB)

B 6 el(M, Ad P) Similarly, following (i), the

Here we follow sections 4.2 and 4.3 of [I]

156

PRICE

~S r , is the linear functional first r-variation of ~ ,

on X(M) given by

~s~(x) :-I. {l~12div x - 4(~(Ve.X , ej), ~(ei, ej))]dV i

Then 6 8 : 0 (@S: : O) is implied by (equivalent to)

being a Yang-Mills connection (being r-stationary).

The total variation ItsS~oll (total r-variation t16s~l 1) of ~ is the largest Borel regular measure on M

determined by the requirement that

H6sJJ(G) : sup{~S (B) : B ( gl(M, Ad P),

supp B c G and I BI < i}

II6<II(G) : sup{6<(x) : x ~ x ( . ) ,

supp X c G and IX[ -< i} .

If G is an open set in M we say

variation (or bounded r-variation) in

c < ~ such that

16s~(B)l _~ c suplBI

has bounded

G if there exists

for all B ( gI(M, Ad P)

similarly for 6S:).

with

Suppose that 116S~ll

supp B c G (and

(IJ6s~Jl) is a Radon

157

PRICE

measure on M . Then ~ has bounded variation

(r-variation) in K for all compact-subsets K of M .

Well known representation theorems ([4] 2.5.12) assert the

existence of a IJ6S~II (II@S:II) measurable section q

(n r) of S(T*M @ Ad P) {S(TM)} , the bundle of unit

length elements of T*M GAd P (TM) , such that

~Sco(B) : fM (B, n)dtl~So~ll

for B ( gI(M, Ad P) , and

: & ( x ,

for X (X(M) If V is the Riemannian volume measure

on M we define, using the theory of symmetrical

derivation (see [4] 2.8.18, 2.9), real-valued V

measurable functions

ll@Smll/V(x) : r+os ll6Smlj (Br( x)] / V(Br(X) )

and similarly for

IL6So~llsing

116S:II/V , such that if

: tl~scoll L { x : 116Scoll/v(x): ~}

(similarly II6S:llsing ) then

116s~tt(G) : fGIt6Scoll / V dV + II@S~IIsing(G)

(similarly II~S:II(G)) whenever G is a Borel subset

158

PRICE

of M

We define the generalised quark current, or

source (r-source) density, H (H r) , of the connection

as

-]l~sJ/v(x)n(x) , H(x)

Hr(x) : II@S:II/V(x)nr(x)

V measurable sections of T*M~ Ad P (TM) These are

and

whenever B

T*M @ Ad P (TM)

(/M IxldlI6S ll <

Note that

@SL0(B) = - fM (B, H)dV + fM (B, n)dll6S011sing ,

~S~(X) : - fM (X, Hr]dV + IM (X, Nr)dlI@S:]Ising

(X) is a Borel measurable section of

and IM IBIdli@SJ < ~

@S (B+V~) : @S (B) for all

infinitesimal gauge transformations ~ ( W 2'2 N W 1'4

(Ad P) When II~S~]J is a Radon measure, as we are

assuming here, this equality remains valid whenever V~

is Borel measurable and fM JV~IdH@S~ li < ~ If

X (X(M) has compact support, supp X , and

= ~ + B ( U 1'2 N U ~ o =~oc ~oc ' X ~ ~ and

VB(X) ( W~ Ad P)) . Also

159

PRICE

IM IxJ~ldll6S~ll = I M Ix-~l IHIdv + IM I•

-< {Isupp x IxJ~12dv}~{l~upp x l~{12dV}~

+ [M IxJ~lall6s~llsing

and similarly ~or f . ]W(X) Idtl6S~II Thus the condition

Isupp x IHI2av ' Isupp x leldll6S~llsing

and fsupp X IvBlall6S~llsing < ~

implies the finiteness of fM IxJeldll6S tl and

fM IVB(X)IdlI@Sm II and hence the equality of

6S~(XJ~+VB(X)) and 6S (XJ~) We then have

6S~(X) = - fM (X, Hr)dV + fM (X, qr)dll~S:llsing

and

6sr(x) = 6S (xJ~) W

- fM (XJe, H)aV + fM (X-m, n)dll~Soltsing

- fM (X, e(., H))dV + fM (X, ~(', n))dlI6S llsing

for all compactly supported X (X(M) satisfying *,

where ~(., A) : (e(ei, ej), A(ej))e i (summation

convention) 6 X(M) for A ( gI(M, Ad P). This implies

160

PRICE

(9)

H r = ~ ( ' , H) , q r = ~ ( . , r l ) / l ~ ( ' , r l ) l

IlSS~llsing : l~( ' , n)[ II~Smllsing

and

on the interior of supp X for any such X .

For the monotonicity formula derived below we

need to assume that ll6S~IlsingIBl(P) ] = 0 and

IHr[ _< Alibi 2 for some finite constant A I

this will be satisfied if

IHI _< A II~] Then, for

By (9)

il6Smllsing(Bz(p) ) = o and

supp X c Bl(P) as before,

~<<x) = - S~ (x, ~r)dv

: SM {lel 2air x 4(~(Ve.X , ej), ~(ei, ej))}dV. !

As in the derivation of (7), with X = ~r ~r '

(X, H r) < IxlA 11~12 : Az Cr l ~ l 2

This leads to

A u o

(i0) e o4-nSBo(x) IAI 2 Ar

+ SBp(X)~B(x)4e o r4-n ~Ja g~i 2

ioP 4-n 2

_< e p SBp(X) I~I

for x (B �89 and 0 < o < p < �89 , where

Ao : cA + A 1 .

161

PRICE

THEOREM I". Let ~ ~ ~1,2 n~o,4(Bl(P) ) have bounded

r-variation in B/(p) . Suppose also that

ll6S~llsing[Bl(P)) : 0 and that the generalised r-source

density H r is bounded as IHrl ~ AII~(~)[ 2 on Bl(P)

for some finite A I . (The latter condition is implied

by a bound on the generalised quark current density H

of IHI ~ AII~(~) I .) Then, for n ~ 4 , (iO) holds

and in particular (3') holds with cA replaced by

cA + A I .

REMARK. Analogous results hold for harmonic maps.

t u = ~t*u for u ( WI'2(M, N) , as before, (8) is

replaced by

E /] (8a) u = du t dt t=O = du(X) (W~ �9

If

The first variation functionals are defined analogously,

we use the notation @E , 6[ r u U

6S 6S r etc. Then

etc. in place of

W I 6[ : '2(u*TN) § U

6[ (Y) =2r dY> = 2fM (du, dY)dV u

6[ r : x(M)§ u

6Er(x) = - fM {[duI2div X- 2(dU(Ve.X ), du6ei))}dV . l

We call the analogs of the generalised quark current

162

PRICE

(r-source) densities the generalised tension (r-tension)

vector fields and denote them T [T r) They are V

measurable sections of u*TN (TM) Defining

(du, A) = (du(ei], A)e i (x(M) for A (F(u*TN) we

have the analog of (9)

(9a) T r : (du, T) , n r = (du , U ) / l ( d u , U) I and

IISE~llsing : t( du, ~)1 116Eulising

on the interior of supp X for any X satisfying the

analog of * Then we have

THEOREM l" (a). For n ~ 2 let u (wl'2(Bl(P), N)

have bounded r-variation in Bl(P) Suppose also that

ll6[~llsing(Bl(P)) = 0 and that the generalised r-tension

vector field T r is bounded as ITrl ~ AlldUl 2 on

BI(p) for some finite A 1 . (The latter condition is

implied by a bound on the generalised tension vector

field �9 of I~1 ~ Al ldul .) men

(lOa) eA~176176 [B (x) 0

du 12 + fBp(X)r~B (x) A r

2e o 2-n Du 2 r I-~-I

AoP 2-n --< e p fB (x) Idul2

P

for x (B�89

Ao = cA + A 1

and 0 < o <_ @ < �89 , where

163

PRICE

REMARK. These monotonicity formulae allow us to define

the densities 9 : M §

~im 4-nSx (ii) @(x) = r+o r r

in the Yang-Mills case, and

~im r2-nE x (lla) 9(x) = r+o r

in the harmonic map case. By (I0) and (10a) these exist

and are non-negative for all x ~ B�89 The positive

density set ~ is defined

= {x : e(x)> 0}

We have the

COROLLARY 3. Assuming the conditions of Theorem i" or

l"a as appropriate, then Hn-4[S~B�89 = 0 in the Yang-

Mills case and Hn-2~inB�89 = 0 in the harmonic map

case, where H k is Hausdorf k-dimensional measure.

REMARK. Schoen and Uhlenbeck ([ii] Theorem 3.1) prove

that for an energy minimising harmonic map the positive

density set coincides with the singular set, also denoted

by them. They prove much stronger results about the

size of this singular set for the energy minimising

case.

PROOF.

S = U S , where S ==Or ==~

0~0

164

= {x : o}

PRICE

A simple covering argument (e.g. as in [ii] Corollary 2.7)

using the monotonicity formulae then proves that

Hn-4(S nB�89 (or Hn-2(S nB�89 as appropriate]

vanishes. U

REMARK. Analogies between the treatment and results for

the harmonic map, Yang-Mills field and minimal surface

problems have been evident in the recent physics and

mathematics literature (e.g. [2, 8, ii, 12, 13, 14, 15,

16, 17]).

REFERENCES:

[i]

[2]

[3]

E43

[5]

[6]

[73

ALLARD, W.K.: On the first variation of a varifold, Ann. of Math. 95, 417-491 (1972)

BOURGUIGNON, J.P., LAWSON, Jr., H.B.: Stability and isolation phenomena for Yang-Mills fields, Commun. Math. Phys. 79, 189-230 (1981)

DE GIORGI, E.: Frontiere orientate di misura minima, Sem. Mat. Scuola Norm. Sup. Pisa (1961)

FEDERER, H.: Geometric measure theory, Springer- Verlag, Berlin (1969)

GARBER, W-D, RUIJSENAARS, S.N.M., SEILER, E., BURNS, D.: On finite action solutions of the nonlinear O-model, Ann. Phys. 119, 305-325 (1979)

HILDEBRANDT, S.: Nonlinear elliptic systems and harmonic mappings, Proceedings of the Beijing Symposium on Differential Geometry and Differential Equations, Beijing, 1980, to appear

LAWSON, Jr., H.B.: Minimal varieties in real and complex geometry, S.M.S. 57, Universite d~ Montreal (1974)

165

PRICE

[8]

[e]

[ lO]

MORREY, Jr., C.B.: The problem of Plateau on a Riemannian manifold, Ann. of Math. 49, 807-851 (1948)

PRICE, P.F., SIMON, L.: Monotonicity formulae for harmonic maps and Yang-Mills fields. Preliminary report. Unpublished

SAMPSON, J.H.: On harmonic mappings, Istit. Naz. Alta Mat., Symp. Mat. 26, 197-210 (1982)

[11]

[12]

[13]

[14]

[15]

[16]

[17]

SCHOEN, R., UHLENBECK, K.: A regularity theory for harmonic maps, J. Diff. Geom. 17, 307-335 (1982)

SIMONS, J.: Minimal varieties in Riemannian manifolds, Ann. of Math. 88, 62-105 (1968)

SIMONS, J.: Gauge fields, a lecture given during the "Japan - United States Seminar on Minimal Submanifolds, including Geodesics:, Tokyo, (1977). (See also [2])

SIU, Y.T., YAU, S.T.: Compact Kahler manifolds of positive bisectional curvature, Invent. Math. 59, 189-204 (1980)

UHLENBECK, K.K.: Removeable singularities in Yang-Mills fields, Commun. Math. Phys. 83, 11-29 (1982)

UHLENBECK, K.K.: Connections with L p bounds on curvature, Commun. Math. Phys. 83, 31-42 (1982)

XIN, Y.L.: Some results on stable harmonic maps, Duke Math, J. 47, 609-613 (1980)

Peter Price Department of Mathematics Institute of Advanced Studies Australian National University P0 Box 4 Canberra, ACT 2600 Australia

(Received December 18, 1983)

166