Stabilities of homothetically shrinking Yang-Mills solitons
aa r X i v : . [ m a t h . DG ] D ec STABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLSSOLITONS
ZHENGXIANG CHEN AND YONGBING ZHANG
Abstract.
In this paper we introduce entropy-stability and F-stability for homoth-etically shrinking Yang-Mills solitons, employing entropy and second variation of F -functional respectively. For a homothetically shrinking soliton which does not descend,we prove that entropy-stability implies F-stability. These stabilities have connectionswith the study of Type-I singularities of the Yang-Mills flow. Two byproducts are alsoincluded: We show that the Yang-Mills flow in dimension four cannot develop a Type-Isingularity; and we obtain a gap theorem for homothetically shrinking solitons. Introduction
In this paper we introduce entropy-stability and F-stability for homothetically shrinking(Yang-Mills) solitons. Let E be a trivial G -vector bundle over R n and of rank r . Here thegauge group G is a Lie subgroup of SO ( r ). A homothetically shrinking soliton, centeredat the space-time point ( x = 0 , t = 1), is a connection A ( x ) on E such that( d ∇ ) ∗ F + 12 i x F = 0 , where F is the curvature of A ( x ), ( d ∇ ) ∗ denotes the formal adjoint of the covariant exteriordifferentiation d ∇ , and i x stands for the interior product by the position vector x .A homothetically shrinking soliton A ( x ) gives rise to a special solution of the Yang-Millsflow. In fact in the exponential gauge of A ( x ), the following A ( x, t ) = A j ( x, t ) dx j := (1 − t ) − A j ((1 − t ) − x ) dx j is a solution to the Yang-Mills flow. On the other hand, homothetically shrinking solitonsare closely related to Type-I singularities of the Yang-Mills flow. Weinkove [22] provedthat Type-I singularities of the Yang-Mills flow are modelled by homothetically shrinkingsolitons whose curvatures do not vanish identically. Examples of homothetically shrinkingsolitons have been found in [10, 22]. In this paper, we restrict ourselves to homotheticallyshrinking solitons which have uniform bounds on |∇ k A ( x ) | for each k ≥
1. In fact,Weinkove showed in [22] that Type-I singularities of the Yang-Mills flow can be modelledby such solitons.Recently, Colding and Minicozzi [8] discovered two functionals for immersed surfaces inEuclidean space, i.e. the F -functional and the entropy. Critical points of both functionalsare self-shrinkers of the mean curvature flow. Colding and Minicozzi introduced entropy-stability and F-stability for self-shrinkers. Inspired by their work, in this paper we aim Mathematics Subject Classification.
Key words and phrases.
Yang-Mills flow, stability, homothetically shrinking soliton.The project is supported by NSFC No. 11201448. to introduce corresponding stabilities for homothetically shrinking Yang-Mills solitons. Infact there are many aspects in common concerning the entropy-stability and F-stability forself-similar solutions to various geometric flows, which includes mean curvature flow, Ricciflow, harmonic map heat flow, and Yang-Mills flow. For the entropy-stability and linearlystability of Ricci solitons, see for instance [4, 6]; for the entropy-stability and F-stabilityof self-similar solutions to the harmonic map heat flow see [23].We begin with the definition of F -functional. Let x be a point in R n and t a positivenumber. The F -functional with respect to ( x , t ), defined on the space of connections on E , is given by(1.1) F x ,t ( A ) = t Z R n | F | (4 πt ) − n e − | x − x | t dx. The functional F x ,t can trace back to the monotonicity formula of the Yang-Mills flow.For the monotonicity formula see [7, 12, 18]. Let A ( x, t ) be a solution to the Yang-Millsflow on E and Φ x ,t ( A ( x, t )) = ( t − t ) Z R n | F | [4 π ( t − t )] − n e − | x − x | t − t ) dx. Along the Yang-Mills flow, Φ x ,t is non-increasing in t . Moreover Φ x ,t is preservedif and only if A ( x,
0) is a homothetically shrinking soliton centered at ( x , t ). Here ahomothetically shrinking soliton centered at ( x , t ) is a connection on E satisfying theequation(1.2) ( d ∇ ) ∗ F + 12 t i x − x F = 0 . The F -functional leads to another characterization of homothetically shrinking solitons:Critical points of F x ,t are exactly homothetically shrinking solitons centered at ( x , t );moreover, ( x , t , A ) is a critical point of the function ( x, t, A )
7→ F x,t ( A ) if and only if A is a homothetically shrinking soliton centered at ( x , t ).The λ -entropy of a connection A ( x ) on the bundle E is defined by(1.3) λ ( A ) = sup x ∈ R n ,t > F x ,t ( A ) . A crucial fact is the following
Proposition 1.1.
Let A ( x, t ) be a solution to the Yang-Mills flow on the bundle E . Thenthe entropy λ ( A ( x, t )) is non-increasing in t . The entropy is a rescaling invariant. More precisely, let A ( x ) be a connection on E and A c a rescaling of A ( x ) given by A ci ( x ) = c − A i ( c − x ), c >
0. Then F cx ,c t ( A c ) = F x ,t ( A ) and hence λ ( A c ) = λ ( A ). In particular the entropy of each time-slice of thehomothetically shrinking Yang-Mills flow, induced from a homothetically shrinking soliton,is preserved. The entropy is also invariant under translations of a connection. Let A ( x )be a connection on E , x ∈ R n a given point, and e A i ( x ) = A i ( x + x ). Then we have F x − x ,t ( e A ) = F x ,t ( A ) and hence λ ( e A ) = λ ( A ).In general the entropy λ ( A ) of a connection A ( x ) is not attained by any F x ,t ( A ).However if A ( x ) is a homothetically shrinking soliton centered at ( x , t ), then λ ( A ) = F x ,t ( A ). In fact we prove the following TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 3
Proposition 1.2.
Let A ( x ) be a homothetically shrinking soliton centered at (0 , suchthat i V F = 0 for any non-zero V ∈ R n . Then the function ( x , t ) F x ,t ( A ) attains itsstrict maximum at (0 , . Note that if i V F = 0 for some non-zero vector V , then A ( x ) can be viewed as aconnection on a G -vector bundle over any hyperplane perpendicular to V and we say A ( x )descends (to V ⊥ ).Entropy-stability and F-stability are defined for homothetically shrinking solitons. Definition 1.1.
A homothetically shrinking soliton A ( x ) is called entropy-stable if it isa local minimum of the entropy, among all perturbations e A ( x ) such that || e A − A || C issufficiently small. Entropy-stability of homothetically shrinking solitons has direct connections with Type-I singularities of the Yang-Mills flow. For example, given an entropy-unstable homothet-ically shrinking soliton A ( x ), by definition we can find a perturbation e A ( x ) of A ( x ) suchthat || e A − A || C is arbitrarily small and has less entropy. Then by comparing the entropy,the Yang-Mills flow starting from e A cannot converge back to a rescaling of A ( x ). More-over, the Yang-Mills flow cannot develop a Type-I singularity modelled by A ( x ), due tothe fact that the entropy is a rescaling invariant.Let A ( x ) be a homothetically shrinking soliton centered at ( x , t ). For a 1-parameterfamily of deformations ( x s , t s , A s ) of ( x , t , A ), let V = dx s ds | s =0 , q = dt s ds | s =0 , θ = dA s ds | s =0 . Definition 1.2. A ( x ) is called F-stable if for any compactly supported θ , there exist areal number q and a vector V such that F ′′ x ,t ( q, V, θ ) := d ds | s =0 F x s ,t s ( A s ) ≥ . Entropy-stability has an apparent connection with the singular behavior of the Yang-Mills flow; however the F-stability is more practical when we are trying to do classification.The classification of entropy-stable homothetically shrinking solitons can be relied on theclassification of F-stable ones. In fact we have the following relation for entropy-stabilityand F-stability.
Theorem 1.3.
Let A ( x ) be a homothetically shrinking soliton such that i V F = 0 for anynon-zero V ∈ R n . If A ( x ) is entropy-stable, then it is F-stable. Let A ( x ) be a homothetically shrinking soliton centered at (0 , Lθ = − [( d ∇ ) ∗ d ∇ θ + R ( θ ) + i x d ∇ θ ] , where R ( θ )( ∂ j ) := [ F ij , θ i ]. For the homothetically shrinking soliton A ( x ), we have(1.5) L ( d ∇ ) ∗ F = ( d ∇ ) ∗ F and Li V F = 12 i V F, ∀ V ∈ R n . The second variation of the F -functional and at A is given by(1.6)12 F ′′ , ( q, V, θ ) = Z R n < − Lθ + 2 q ( d ∇ ) ∗ F − i V F, θ > Gdx − Z R n ( q | ( d ∇ ) ∗ F | + 12 | i V F | ) Gdx,
ZHENGXIANG CHEN AND YONGBING ZHANG where G ( x ) = (4 π ) − n exp( − | x | ). Denote the space of θ satisfying Lθ = − λθ by E λ . Wehave the following characterization for F-stability. Theorem 1.4. A ( x ) is F-stable if and only if the following properties are satisfied • E − = { c ( d ∇ ) ∗ F, c ∈ R } ; • E − = { i V F, V ∈ R n } ; • E λ = { } , for any λ < and λ = − , − . Theorem 1.4 amounts to say that A ( x ) is F-stable if and only if L is non-negative def-inite modulo the vector space spanned by ( d ∇ ) ∗ F and i V F . This is actually the reflectionof the invariance property of the F -functional and the entropy under rescalings and trans-lations. Since Colding-Minicozzi’s work [8], classification problem of F-stable self-shrinkersof the mean curvature flow has drawn much attention, see for instance [1, 2, 16, 17].We have two simple byproducts regarding homothetically shrinking solitons. We showthe non-existence of homothetically shrinking solitons in dimensions four and lower, anda gap theorem. Let A ( x ) be a homothetically shrinking soliton centered at (0 , Z R n | x | | F | G ( x ) dx = 2( n − Z R n | F | G ( x ) dx. It immediately implies the following
Proposition 1.5.
When n = 2 , , or , there exists no homothetically shrinking solitonsuch that | F | is uniformly bounded and not identically zero. R˚ade [19] proved that the Yang-Mills flow, over a compact Riemannian manifold ofdimension n = 2 or 3, exists for all time and converges to a Yang-Mills connection. Howeverif the base manifold has dimension five or above, Naito [18] showed that the Yang-Mills flowcan develop a singularity in finite time, see also [11]. It is unclear yet whether the Yang-Mills flow over a four-dimensional manifold develops a singularity in finite time. For partialresults in this dimension, see for instance [13, 20, 21] , the remarkable monographs [9] andthe references therein . Together with Weinkove’s blowup analysis for Type-I singularitiesof the Yang-Mills flow, Proposition 1.5 shows that the Yang-Mills flow cannot develop asingularity of Type-I. This was actually a known fact, see for instance [12].Gap theorems for Yang-Mills connections over spheres was considered in [3]. Gap the-orems for various kinds of self-similar solutions have also been obtained, see for instance[5, 15, 23]. By (1.5), we have the following gap result for homothetically shrinking solitons. Theorem 1.6.
Let A ( x ) be a homothetically shrinking soliton centered at (0 , . If | F | < n n − , then ( E, A ) is flat. The paper is organized as follows: in the next section we review some background,with emphasis on homothetically shrinking solitons and Weinkove’s blowup analysis forType-I singularities of the Yang-Mills flow. In Section 3, we consider the F -functionaland its first variation. Section 4 is devoted to the calculation of the second variationof the F -functional, i.e. (1.6). In Section 5, we study the F-stability of homotheticallyshrinking solitons and prove Theorem 1.4 and Theorem 1.6. In Section 6, we introduce the λ -entropy and prove Proposition 1.2. In the last section, we prove that entropy-stabilityimplies F-stability, i.e. Theorem 1.3. TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 5
We would like to point out that although we assume, for simplicity, that the homothet-ically shrinking solitons have uniform bounds on |∇ k A | , our statements except Theorem1.3 are still straightforwardly valid if |∇ k A | has polynomial growth. Many results in thispaper have also been obtained by Kelleher and Streets [14] independently. Preliminaries
In this section we briefly introduce the Yang-Mills flow and its singularity. We shallintroduce the blowup analysis for Type-I singularities, which was carried out by Weinkove[22]. It leads to the main object in this paper, i.e. homothetically shrinking soliton.Let (
M, g ) be a closed n -dimensional Riemannian manifold. Let G be a compact Liegroup and P ( M, G ) a principle bundle over M with the structure group G . We fix a G -vector bundle E M = P ( M, G ) × ρ R r , associated to P ( M, G ) via a faithful representation ρ : G → SO ( r ). Let g denote the Lie algebra of G . A connection on E M is locally a g -valued 1-form. Using Latin letters for the manifold indices, one may write a connection A in the form of A = A i dx i , where A i ∈ so ( r ). Using Greek letters for the bundle indices,one may also write A = A αiβ dx i . The curvature of the connection A is locally a g -valued2-form F = 12 F ij dx i ∧ dx j = 12 F αijβ dx i ∧ dx j , and F ij = ∂ i A j − ∂ j A i + [ A i , A j ] . The Yang-Mills functional, defined on the space of connections, is given by(2.1)
Y M ( A ) = 12 Z M | F | dµ g , where | F | = 12 g ik g jl < F ij , F kl > = 12 g ik g jl F αijβ F αklβ , and F αijβ = ∂ i A αjβ − ∂ j A αiβ + A αiγ A γjβ − A αjγ A γiβ . Let ∇ denote the covariant differentiation on Γ( E M ) associated to the connection A , andalso the covariant differentiation on g -valued p -forms induced by A and the Levi-Civitaconnection of ( M, g ). Curvature F satisfies the Bianchi identity d ∇ F = 0, where d ∇ denotes the covariant exterior differentiation. Let ( d ∇ ) ∗ denote the formal adjoint of d ∇ . A connection A is a critical point of the Yang-Mills functional, called a Yang-Millsconnection, if and only if it is a solution of the Yang-Mills equation ( d ∇ ) ∗ F = 0. TheYang-Mills equation can also be written as ∇ p F αpjβ = 0 . In normal coordinates of (
M, g ), we have ∇ p F αpjβ = ∂ p F αpjβ + A αpγ F γpjβ − F αpjγ A γpβ . As the L -gradient flow of the Yang-Mills functional, the Yang-Mills flow is defined by(2.2) dAdt = − ( d ∇ ) ∗ F. ZHENGXIANG CHEN AND YONGBING ZHANG
Assume A ( x, t ) is a smooth solution to the Yang-Mills flow for 0 ≤ t < T and as t → T the curvature blows up, i.e. lim sup t → T max x ∈ M | F ( x, t ) | = ∞ . If there exists a positiveconstant C such that(2.3) | F ( x, t ) | ≤ CT − t , one says that the Yang-Mills flow develops a Type-I singularity, or a rapidly formingsingularity. Otherwise one says that the Yang-Mills flow develops a Type-II singularity.If (2.3) is satisfied and x is a point such that lim sup t → T | F ( x , t ) | = ∞ , we call ( x , T ) aType-I singularity.Let A ( x, t ) be a smooth solution to the Yang-Mills flow and ( x , T ) a Type-I singularity.We now follow [22] introducing the blowup procedure around ( x , T ). Let B r ( x ) be asmall geodesic ball centered at x and of radius r over which E M is trivial. For simplicitywe identify B r ( x ) with the ball B r (0) in R n . Let λ i be a sequence of positive numberstending to zero. For each i , one gets a Yang-Mills flow A λ i ( y, s ) by setting(2.4) A λ i ( y, s ) = λ i A p ( λ i y, T + λ i s ) dy p , y ∈ B r/λ i (0) , s ∈ [ − λ − i T, . (An alternative way of obtaining a sequence of blowups of A ( x, t ) is to rescale the metricaround the singular point x .) Let x = λ i y and t = T + λ i s . By the assumption (2.3), thecurvature of A λ i satisfies | F λ i ( y, s ) | = λ i | F ( x, t ) | = | s | − ( T − t ) | F ( x, t ) | ≤ C | s | − . Let h = h αβ be a gauge transformation which acts on connections by h ∗ ∇ = h − ◦ ∇ ◦ h, or equivalently, h ∗ A = h − dh + h − Ah.
Note that gauge transformations preserve Yang-Mills flows. Hence h ∗ A λ i ( y, s ) defines asolution to the Yang-Mills flow. Weinkove [22] proved the following Theorem 2.1.
Let ( x , T ) be a Type-I singularity of the Yang-Mills flow A ( x, t ) over M .Then there exists a sequence of blowups A λ i ( y, s ) defined by (2.4) and a sequence of gaugetransformations h i such that h ∗ i A λ i ( y, s ) converges smoothly on any compact set to a flow e A ( y, s ) . Here e A ( y, s ) , defined on a trivial G -vector bundle over R n × ( −∞ , , is a solutionto the Yang-Mills flow, which has non-zero curvature and satisfies (2.5) e ∇ p e F pj − | s | y p e F pj = 0 . In Theorem 2.1, h i are chosen as suitable Coulomb gauge transformations so that forany s < k ≥ |∇ k h ∗ i A λ i | is uniformly bounded. The bounds do not depend on i .Hence for any s < k ≥ |∇ k e A | is uniformly bounded.A solution A ( y, s ) to the Yang-Mills flow, defined on a trivial bundle over R n × ( −∞ , A αiβ ( y, s ) = 1 p | s | A αiβ ( y p | s | , − TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 7 for any y ∈ R n and s <
0, for more details see [22]. The limiting Yang-Mills flow e A ( y, s ) isactually a homothetically shrinking soliton. In fact via an exponential gauge for e A ( y, s ),in which y p e A αpβ = 0, (2.5) and (2.6) are equivalent for the Yang-Mills flow e A ( y, s ).One of the main ingredients of Theorem 2.1 is the monotonicity formula for the Yang-Mills flow, see [7, 12, 18]. In the simplest case that A ( x, t ) is a solution to the Yang-Millsflow over R n , one can define(2.7) Φ x ,t ( A ( x, t )) = ( t − t ) Z R n | F ( x, t ) | G x ,t ( x, t ) dx, here t > , t ∈ [0 , min { T, t } ), and G x ,t ( x, t ) = [4 π ( t − t )] − n exp( − | x − x | t − t ) ) is thebackward heat kernel. The monotonicity formula of the Yang-Mills flow reads(2.8) ddt Φ x ,t ( A ( x, t )) = − t − t ) Z R n |∇ p F pj − t − t ) ( x − x ) p F pj | G x ,t ( x, t ) dx. The monotonicity Φ x ,t is non-increasing in t , and is preserved if and only if(2.9) ∇ p F pj − t − t ) ( x − x ) p F pj = 0 . For the limiting Yang-Mills flow e A ( y, s ) obtained in Theorem 2.1 and any ( x , t ) ∈ R n × (0 , + ∞ ), one can translate it into(2.10) A ( x, t ) = A p ( x, t ) dx p = e A p ( x − x , t − t ) dx p , then A ( x, t ) is a solution to the Yang-Mills flow and (2.9) is satisfied. On the other handif a connection A ( x ) on a trivial G -vector bundle over R n satisfying ∇ p F pj − t ( x − x ) p F pj = 0 , then, in the exponential gauge for A ( x ), i.e. a gauge such that ( x − x ) p A p ( x ) = 0, theflow of connections given by A p ( x, t ) := r t t − t A p ( x + r t t − t ( x − x ))is a solution to the Yang-Mills flow which satisfies (2 . e A ( y, s ), homothetically shrinking solitons A ( x ) and homothetically shrinkingYang-Mills flows are the same thing.From now on we assume that E is a trivial G -vector bundle over R n . Definition 2.1.
A connection A ( x ) on E is called a homothetically shrinking solitoncentered at ( x , t ) if it satisfies (2.11) ∇ p F pj − t ( x − x ) p F pj = 0 . Let A ( x ) be a homothetically shrinking soliton centered at ( x , t ) and A ( x, t ) the Yang-Mills flow initiating from A ( x ). In an exponential gauge such that ( x − x ) p A p ( x, t ) = 0,we have for any λ > t < t that A j ( x, t ) = λA j ( λ ( x − x ) + x , λ ( t − t ) + t ). ZHENGXIANG CHEN AND YONGBING ZHANG F -functional and its first variation In this section we define the F -functional of connections on the trivial G -vector bundle E over R n . Homothetically shrinking solitons are critical points of the F -functional.We shall prove necessary integral identities for homothetically shrinking solitons. As acorollary of one of these identities, we give a proof of the fact that the Yang-Mills flow indimension four cannot develop a Type-I singularity.For convenience, we set two g -valued 1-forms J and X , respectively, by J := ∇ p F pj dx j , X := i x − x F = ( x − x ) p F pj dx j . According to (2.11), A ( x ) is a homothetically shrinking soliton centered at ( x , t ) if andonly if J = 12 t X. We also set(3.1) S x ,t = { A ( x ) : A is a homothetically shrinking soliton with sup |∇ k A | < ∞ , ∀ k ≥ } . Note that for any k ≥
1, any time-slice e A ( · , s ) in Theorem 2.1 satisfies sup |∇ k e A ( · , s ) | < ∞ . Definition 3.1.
For any x ∈ R n , t > , the F -functional with respect to ( x , t ) isdefined by (3.2) F x ,t ( A ) = t Z R n | F | (4 πt ) − n e − | x − x | t dx. We now compute the first variation of the F -functional. Consider a differentiable 1-parameter family ( x s , t s , A s ), where A = A . Denote˙ t s = dds t s , ˙ x s = dds x s , θ s = dds A s , and G s ( x ) = (4 πt s ) − n e − | x − xs | ts . Proposition 3.1.
Assume |∇ k A s | < ∞ for any k ≥ and R R n ( | θ s | + |∇ θ s | ) G s dx < ∞ . The first variation of the F -functional is given by dds F x s ,t s ( A s ) = Z R n ˙ t s ( 4 − n t s + 14 | x − x s | ) | F s | G s ( x ) dx + Z R n t s < ˙ x s , x − x s > | F s | G s ( x ) dx − Z R n t s < θ s , J s − X s t s > G s ( x ) dx. (3.3) Proof.
Note that ∂∂s G s ( x ) = ( − n t s t s + ˙ t s | x − x s | t s + < ˙ x s , x − x s > t s ) G s ( x ) , and ∂∂s | F s | = F αijβ ( ∇ i θ αjβ − ∇ j θ αiβ ) , TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 9 so we have dds F x s ,t s ( A s ) = Z R n t s ˙ t s | F s | G s ( x ) dx + Z R n t s F αijβ ( ∇ i θ αjβ − ∇ j θ αiβ ) G s ( x ) dx + Z R n t s | F s | ( − n t s t s + ˙ t s | x − x s | t s + < ˙ x s , x − x s > t s ) G s ( x ) dx = Z R n t s ˙ t s | F s | G s ( x ) dx + Z R n t s F αijβ ∇ i θ αjβ G s ( x ) dx + Z R n t s | F s | ( − n t s t s + ˙ t s | x − x s | t s + < ˙ x s , x − x s > t s ) G s ( x ) dx. Let η ( x ) be a cutoff function on R n . By integration by parts, we have Z R n t s F αijβ ∇ i θ αjβ G s ( x ) η ( x ) dx = Z R n − t s θ αjβ [ ∇ i F αijβ G s η + F αijβ ∂ i ( G s ) η + F αijβ G s ∂ i η ] dx = Z R n − t s θ αjβ [ ∇ i F αijβ η − ( x − x s ) i t s F αijβ η + F αijβ ∂ i η ] G s dx. (3.4)Let η l ( x ) = 1 for | x | ≤ l , and cut off to zero linearly on B l +1 \ B l . Taking η = η l in (3.4)and applying the Lebesgue’s dominated convergence theorem, we get(3.5) Z R n t s F αijβ ∇ i θ αjβ G s ( x ) dx = Z R n θ αjβ [ − t s ∇ i F αijβ + t s ( x − x s ) i F αijβ ] G s dx. Hence we get dds F x s ,t s ( A s ) = Z R n t s ˙ t s | F s | G s ( x ) dx + Z R n θ αjβ [ − t s ∇ i F αijβ + t s ( x − x s ) i F αijβ ] G s ( x ) dx + Z R n t s | F s | ( − n t s t s + ˙ t s | x − x s | t s + < ˙ x s , x − x s > t s ) G s ( x ) dx. = Z R n ˙ t s ( 4 − n t s + 14 | x − x s | ) | F s | G s ( x ) dx + Z R n t s < ˙ x s , x − x s > | F s | G s ( x ) dx − Z R n t s < θ s , J s − X s t s > G s ( x ) dx. (cid:3) From Proposition 3.1, we have the following
Corollary 3.1.
A connection A ( x ) is a critical point of F x ,t if and only if A ( x ) is ahomothetically shrinking soliton centered at ( x , t ) . We shall check that ( A ( x ) , x , t ) is a critical point of the F -functional ( e A, x, t ) F x,t ( e A ) if and only if A ( x ) is a homothetically shrinking soliton centered at ( x , t ). To check this we need some identities for homothetically shrinking solitons. We also needsuch identities in the calculation of the second variation of the F -functional in the nextsection. Denote G ( x ) = (4 πt ) − n e − | x − x | t . Lemma 3.2.
Let A ( x ) be a homothetically shrinking soliton centered at ( x , t ) and sup | F ( x ) | < ∞ . Let ϕ = ϕ p ∂ p be a vector field on R n such that | ϕ | is a polynomialin | x − x | , and V a vector in R n . Then we have Z R n ϕ p ( x − x ) p | F | G ( x ) dx = Z R n [2 t ∂ p ( ϕ p ) | F | − t ∂ i ϕ p F αpjβ F αijβ ] G ( x ) dx. In particular, (a) R R n | x − x | | F | G ( x ) dx = R R n n − t | F | G ( x ) dx ;(b) R R n ( x − x ) k | F | G ( x ) dx = 0;(c) R R n | x − x | | F | G ( x ) dx = R R n [4( n − n − t | F | − t | J | ] Gdx ;(d) R R n | x − x | < V, x − x > | F | G ( x ) dx = 0;(e) R R n < x − x , V > | F | Gdx = R R n (2 t | V | | F | − t < V i F ij , V p F pj > ) Gdx .Proof.
Let η ( x ) be a cutoff function on R n . By integration by parts, we get Z R n ϕ p ( x − x ) p | F | G ( x ) η ( x ) dx = Z R n − t ϕ p | F | ∂ p G ( x ) η ( x ) dx = Z R n t [ ∂ p ( ϕ p ) | F | η + ϕ p ∂ p ( | F | ) η + ϕ p | F | ∂ p η ] G ( x ) dx. By integration by parts we have Z R n t ϕ p F αpjβ J αjβ Gηdx = Z R n t ϕ p [ ∇ i ( F αpjβ F αijβ ) − ∇ i F αpjβ F αijβ ] Gηdx = Z R n − t F αpjβ F αijβ [ ∂ i ϕ p − ( x − x ) i t ϕ p ] Gηdx − Z R n t ϕ p ( ∇ i F αpjβ F αijβ + ∇ j F αipβ F αijβ ) Gηdx − Z R n t ϕ p F αpjβ F αijβ G∂ i ηdx. It then follows from the Bianchi identity that Z R n t ϕ p F αpjβ J αjβ Gηdx = Z R n − t F αpjβ F αijβ [ ∂ i ϕ p − ( x − x ) i t ϕ p ] Gηdx − Z R n t ϕ p ∇ p F αijβ F αijβ Gηdx − Z R n t ϕ p F αpjβ F αijβ G∂ i ηdx = Z R n − t F αpjβ F αijβ [ ∂ i ϕ p − ( x − x ) i t ϕ p ] Gηdx − Z R n t ϕ p ∂ p ( | F | ) Gηdx − Z R n t ϕ p F αpjβ F αijβ G∂ i ηdx, TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 11 i.e. Z R n t ϕ p ∂ p ( | F | ) Gηdx = − Z R n t ϕ p F αpjβ J αjβ Gηdx − Z R n t F αpjβ F αijβ [ ∂ i ϕ p − ( x − x ) i t ϕ p ] Gηdx − Z R n t ϕ p F αpjβ F αijβ G∂ i ηdx. Thus we have Z R n ϕ p ( x − x ) p | F | G ( x ) η ( x ) dx = Z R n t [ ∂ p ( ϕ p ) | F | η + ϕ p ∂ p ( | F | ) η + ϕ p | F | ∂ p η ] G ( x ) dx = Z R n t ∂ p ( ϕ p ) | F | Gηdx − Z R n t ϕ p F αpjβ J αjβ Gηdx − Z R n t F αpjβ F αijβ [ ∂ i ϕ p − ( x − x ) i t ϕ p ] Gηdx − Z R n t ϕ p F αpjβ F αijβ G∂ i ηdx + Z R n t ϕ p | F | G∂ p ηdx = Z R n [2 t ∂ p ( ϕ p ) | F | − t ∂ i ϕ p F αpjβ F αijβ ] Gηdx − Z R n t ϕ p F αpjβ ( J αjβ − t X αjβ ) Gηdx − Z R n t ϕ p F αpjβ F αijβ G∂ i ηdx + Z R n t ϕ p | F | G∂ p ηdx. Therefore for a homothetically shrinking soliton centered at ( x , t ), Z R n ϕ p ( x − x ) p | F | Gηdx = Z R n [2 t ∂ p ( ϕ p ) | F | − t ∂ i ϕ p F αpjβ F αijβ ] Gηdx − Z R n t ϕ p F αpjβ F αijβ G∂ i ηdx + Z R n t ϕ p | F | G∂ p ηdx. (3.6)Applying to (3.6) with η ( x ) = η l ( x ), where η l ( x ) = 1 for | x | ≤ l and is cut off to zerolinearly on B l +1 \ B l , we get(3.7) Z R n ϕ p ( x − x ) p | F | Gdx = Z R n [2 t ∂ p ( ϕ p ) | F | − t ∂ i ϕ p F αpjβ F αijβ ] Gdx.
Taking ϕ p = ( x − x ) p , by (3.7) we get Z R n | x − x | | F | G ( x ) dx = Z R n n − t | F | G ( x ) dx. Taking ϕ p = δ pk , by (3.7) we get for any k = 1 , · · · , n , Z R n ( x − x ) k | F | G ( x ) dx = 0 . Taking ϕ p = | x − x | ( x − x ) p , by (3.7) and (a) we get Z R n | x − x | | F | G ( x ) dx = Z R n [2 t ( n + 2) | x − x | | F | − t | x − x | | F | − t | X | ] Gdx = Z R n [4( n − n − t | F | − t | J | ] Gdx.
Taking ϕ p = | x − x | V p , by (3.7) and (b) we get Z R n | x − x | < V, x − x > | F | G ( x ) dx = Z R n − t < J j , V p F pj > Gdx. On the other hand taking ϕ p = < V, x − x > ( x − x ) p , by (3.7) and (b) we get Z R n | x − x | < V, x − x > | F | G ( x ) dx = Z R n − t < J j , V i F ij > Gdx. Thus we have Z R n | x − x | < V, x − x > | F | G ( x ) dx = Z R n < J j , V p F pj > Gdx = 0 . Taking ϕ p = < V, x − x > V p , by (3.7) we get Z R n < x − x , V > | F | Gdx = Z R n (2 t | V | | F | − t < V i F ij , V p F pj > ) Gdx. (cid:3)
By the first variation formula (3.3), (a) and (b) of Lemma 3.2 we get the following
Corollary 3.2. ( A ( x ) , x , t ) is a critical point of the F -functional if and only if A ( x ) isa homothetically shrinking soliton centered at ( x , t ) . Corollary 3.3.
When n = 2 , , or , there exists no homothetically shrinking soliton suchthat | F | is uniformly bounded and not identically zero. In particular in dimension four,the Yang-Mills flow on E M cannot develop a singularity of Type-I.Proof. The first part follows from Lemma 3.2 (a). By Weinkove’s result [22], see also Sec-tion 2, at a Type-I singularity of a Yang-Mills flow one can obtain a homothetically shrink-ing soliton on a trivial G -vector bundle over R n whose curvature is uniformly bounded andnon-zero. Therefore in dimension four if a Type-I singularity occurs, it would contradictwith the non-existence of such a homothetically shrinking soliton. (cid:3) Second variation of F -functional We now compute the second variation of the F -functional at a homothetically shrinkingsoliton A ( x ). Let d ∇ denote the covariant exterior differentiation on g -valued forms and( d ∇ ) ∗ denote the formal adjoint of d ∇ . For a g -valued 1-form θ , let(4.1) R ( θ j ) = R ( θ )( ∂ j ) := [ F ij , θ i ] , and(4.2) Lθ := − t [( d ∇ ) ∗ d ∇ θ + R ( θ ) + i t ( x − x ) d ∇ θ ] . TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 13
We also introduce the space(4.3) W , G := { θ : Z R n ( | θ | + |∇ θ | + | Lθ | ) G ( x ) dx < ∞} . Denote ˙ t s | s =0 = q, ˙ x s | s =0 = V, θ = dds | s =0 A s , F ′′ x ,t ( q, V, θ ) = d ds | s =0 F x s ,t s ( A s ) . Proposition 4.1.
Let A ( x ) be a homothetically shrinking soliton in S x ,t , see (3.1). Thenfor any θ ∈ W , G , we have (4.4) 12 t F ′′ x ,t ( q, V, θ ) = Z R n < − Lθ − qJ − i V F, θ > Gdx − Z R n ( q | J | + 12 | i V F | ) Gdx.
Proof.
Recall that dds F x s ,t s ( A s ) = Z R n ˙ t s ( 4 − n t s + 14 | x − x s | ) | F s | G s ( x ) dx + Z R n t s < ˙ x s , x − x s > | F s | G s ( x ) dx − Z R n t s < J s − X s t s , θ s > G s ( x ) dx. By the assumption that A ( x ) ∈ S x ,t and Lemma 3.2 (a, b), we have F ′′ x ,t ( q, V, θ ) = Z R n [ q ( 4 − n q − < x − x , V > ) + 12 t < V, − V > ] | F | Gdx + Z R n [ q ( 4 − n t + 14 | x − x | ) + 12 t < V, x − x > ] ∂ | F s | ∂s | s =0 Gdx + Z R n [ q ( 4 − n t + 14 | x − x | ) + 12 t < V, x − x > ] | F | ∂G s ∂s | s =0 dx − Z R n t < ∂∂s | s =0 ( J s − X s t s ) , θ > Gdx. Note that ∂ | F s | ∂s | s =0 = F αijβ ( ∇ i θ αjβ − ∇ j θ αiβ ) = 2 F αijβ ∇ i θ αjβ ,∂G s ∂s | s =0 = ( − n qt + q | x − x | t + < V, x − x > t ) G ( x ) ,∂∂s | s =0 J αjβ = ∇ p ∇ p θ αjβ − ∇ p ∇ j θ αpβ + θ αpγ F γpjβ − F αpjγ θ γpβ ,∂∂s | s =0 ( − t s X αjβ ) = q t X αjβ + 12 t V k F αkjβ − t ( x − x ) k ( ∇ k θ αjβ − ∇ j θ αkβ ) . Thus we get F ′′ x ,t ( q, V, θ ) = Z R n [ q ( 4 − n q − < x − x , V > ) − t | V | ] | F | Gdx + Z R n [ q ( 4 − n t + 14 | x − x | ) + 12 t < V, x − x > ]2 F αijβ ∇ i θ αjβ Gdx + Z R n [ q ( 4 − n t + 14 | x − x | ) + 12 t < V, x − x > ] | F | × ( − n qt + q | x − x | t + < V, x − x > t ) Gdx − Z R n t [ ∇ p ( ∇ p θ αjβ − ∇ j θ αpβ ) + θ αpγ F γpjβ − F αpjγ θ γpβ ] θ αjβ Gdx − Z R n t [ q t X αjβ + 12 t V k F αkjβ − t ( x − x ) k ( ∇ k θ αjβ − ∇ j θ αkβ )] θ αjβ Gdx.
By integration by parts, we have Z R n [ q ( 4 − n t + 14 | x − x | ) + 12 t < V, x − x > ]2 F αijβ ∇ i θ αjβ Gdx = Z R n − q ( 4 − n t + 14 | x − x | ) + 12 t < V, x − x > ] < J − t X, θ > Gdx − Z R n
2[ 12 q ( x − x ) i + 12 t V i ] F αijβ θ αjβ Gdx = Z R n [ − q ( x − x ) i − t V i ] F αijβ θ αjβ Gdx.
Then by using Lemma 3.2, we have F ′′ x ,t ( q, V, θ ) = Z R n [ 4 − n q − t | V | ] | F | Gdx + Z R n [ − q ( x − x ) i − t V i ] F αijβ θ αjβ Gdx + Z R n [ n − q | F | − t q | J | ] Gdx + Z R n
14 (2 t | V | | F | − t < V i F ij , V p F pj > ) Gdx − Z R n t [ ∇ p ( ∇ p θ αjβ − ∇ j θ αpβ ) + θ αpγ F γpjβ − F αpjγ θ γpβ ] θ αjβ Gdx − Z R n [ q ( x − x ) i + t V i ] F αijβ θ αjβ Gdx + Z R n t ( x − x ) k ( ∇ k θ αjβ − ∇ j θ αkβ ) θ αjβ Gdx.
TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 15
Thus, F ′′ x ,t ( q, V, θ ) = Z R n [ − q ( x − x ) i − t V i ] F αijβ θ αjβ Gdx − Z R n t q | J | Gdx − Z R n t < V i F ij , V p F pj > ) Gdx − Z R n t [ ∇ p ( ∇ p θ αjβ − ∇ j θ αpβ ) + θ αpγ F γpjβ − F αpjγ θ γpβ ] θ αjβ Gdx + Z R n t ( x − x ) k ( ∇ k θ αjβ − ∇ j θ αkβ ) θ αjβ Gdx.
Note that ( d ∇ ) ∗ d ∇ θ j = −∇ p ( ∇ p θ j − ∇ j θ p ) , R ( θ j ) = [ F pj , θ p ] = F pj θ p − θ p F pj ,i x − x d ∇ θ j = ( x − x ) k ( ∇ k θ j − ∇ j θ k ) , so we have F ′′ x ,t ( q, V, θ ) = Z R n t < ( d ∇ ) ∗ d ∇ θ j + R ( θ j ) + i t ( x − x ) d ∇ θ j , θ j > Gdx − Z R n t < qJ j + V i F ij , θ j > Gdx − t Z R n ( q | J | + 12 | i V F | ) Gdx.
Let L = − t [( d ∇ ) ∗ d ∇ + R + i t ( x − x ) d ∇ ] , then we have12 t F ′′ x ,t ( q, V, θ ) = Z R n < − Lθ − qJ − i V F, θ > Gdx − Z R n ( q | J | + 12 | i V F | ) Gdx. (cid:3) F-stability and its characterization
In this section we define the F-stability for homothetically shrinking solitons in S x ,t .The operator L admits eigenfields J and i V F of eigenvalues − − , respectively.F-stability is equivalent to the semi-positiveness of L modulo the vector space spanned by J and i V F . Let C ∞ (Ω ⊗ g ), or simply C ∞ , denote the space of g -valued 1-forms withcompact supports on R n . The space C ∞ is dense in W , G . Definition 5.1.
A homothetically shrinking soliton A ∈ S x ,t is called F-stable if for any θ in C ∞ , or equivalently in W , G , there exist a real number q and a vector V such that F ′′ x ,t ( q, V, θ ) ≥ . Given a homothetically shrinking soliton A ∈ S x ,t with an exponential gauge, therescaling e A i ( x ) = √ t A i ( √ t x + x )is a homothetically shrinking soliton in S , . Without loss of generality, in the remainingof this section we let x = 0 and t = 1. Then G ( x ) = (4 π ) − n e − | x | and Lθ = − [( d ∇ ) ∗ d ∇ θ + R ( θ ) + i x d ∇ θ ] . The operator L is self-adjoint in the following sense: for any θ, η ∈ W , G ,(5.1) Z R n < Lθ, η > Gdx = − Z R n [ < d ∇ θ, d ∇ η > + < R ( θ ) , η > ] Gdx = Z R n < θ, Lη > Gdx. A g -valued 1-form θ ∈ W , G is called an eigenfield of L and of eigenvalue λ if Lθ = − λθ .We denote the eigenfield space of eigenvalue λ by E λ . Proposition 5.1.
Let A be a homothetically shrinking soliton in S , . Then (5.2) LJ = J, and (5.3) L ( i V F ) = 12 i V F, ∀ V ∈ R n . Proof.
Note that J j = ∇ p F pj = 12 x p F pj ,L = − ( d ∇ ) ∗ d ∇ − R − i x d ∇ , and LJ j = ∇ p ∇ p J j − ∇ p ∇ j J p − [ F pj , J p ] −
12 ( d ∇ J )( x p ∂ p , ∂ j )= ∇ p ∇ p J j − ∇ p ∇ j J p − [ F pj , J p ] − x p ( ∇ p J j − ∇ j J p ) . We have ∇ p J j = ∇ p ( 12 x q F qj ) = 12 F pj + 12 x q ∇ p F qj , then ∇ p ∇ j J p = ∇ p ( − F pj + 12 x q ∇ j F qp ) = − J j − x q ∇ p ∇ j F pq , TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 17 and by using the Bianchi identity and the Ricci formula, we get ∇ p ∇ p J j = ∇ p F pj + 12 x q ∇ p ∇ p F qj = ∇ p F pj + 12 x q ∇ p ( −∇ q F jp − ∇ j F pq )= ∇ p F pj − x q ( ∇ q ∇ p F jp + F pq F jp − F jp F pq ) − x q ∇ p ∇ j F pq = J j + 12 x q ∇ q J j + [ J p , F jp ] − x q ∇ p ∇ j F pq . Hence LJ j = 32 J j + 12 x p ∇ j J p . The identity (5.2) then follows from12 x p ∇ j J p = 12 ∇ j ( x p J p ) − J j = 12 ∇ j ( x p x q F qp ) − J j = − J j . We now prove (5.3). By using the Bianchi identity and the Ricci formula, we get ∇ p ∇ p ( V q F qj ) = V q ∇ p ( −∇ q F jp − ∇ j F pq )= − V q ( ∇ q ∇ p F jp + F pq F jp − F jp F pq ) − V q ∇ p ∇ j F pq = V q ∇ q ( 12 x p F pj ) + [ V q F qp , F jp ] + ∇ p ∇ j ( V q F qp ) , hence L ( V q F qj ) = ∇ p ∇ p ( V q F qj ) − ∇ p ∇ j ( V q F qp ) − [ F pj , V q F qp ] − x p [ ∇ p ( V q F qj ) − ∇ j ( V q F qp )]= V q ∇ q ( 12 x p F pj ) − x p [ ∇ p ( V q F qj ) − ∇ j ( V q F qp )]= 12 V q F qj + 12 x p V q ( ∇ q F pj + ∇ p F jq + ∇ j F qp )= 12 V q F qj . (cid:3) Corollary 5.1.
Let A be a homothetically shrinking soliton in S , . If | F | < n n − , then ( E, A ) is flat.Proof. Note that J j = ∇ p F pj = x p F pj . By integration by parts, we have Z R n < ( d ∇ ) ∗ d ∇ J + i x d ∇ J, J > Gdx = Z R n | d ∇ J | Gdx.
On the other hand by (5.2), we have Z R n < ( d ∇ ) ∗ d ∇ J + i x d ∇ J, J > Gdx = Z R n < − LJ − R ( J ) , J > Gdx = − Z R n | J | Gdx − Z R n < [ F ij , J i ] , J j > Gdx. For any
B, C ∈ so ( r ), we have | [ B, C ] | ≤ | B || C | , see Lemma 2.30 in [3]. Hence | < [ F ij , J i ] , J j > | ≤ | F ij || J i || J j | = 2 X i 12 ( | J | − X k | J k | ) ≤ | F | r 12 (1 − n ) | J | = r n − n | F || J | and Z R n | d ∇ J | Gdx ≤ Z R n ( r n − n | F | − | J | Gdx. If | F | < n n − , one then gets J = 0. Note that if A ∈ S , has J = 0, then for any t > J = t X . Hence Lemma 3.2 (a) holds for any t > F vanishes. (cid:3) Theorem 5.2. Let A be a homothetically shrinking soliton in S , . Then it is F-stable ifand only if the following properties are satisfied (1) E − = { cJ, c ∈ R } ; (2) E − = { i V F, V ∈ R n } ; (3) E λ = { } , for any λ < and λ = − , − . Proof. Let θ be a g -value 1-form in W , G of the form θ = aJ + i W F + e θ, a ∈ R , W ∈ R n and satisfying Z R n < e θ, J > Gdx = Z R n < e θ, i V F > Gdx = 0 , ∀ V ∈ R n . Then it follows from Proposition 4.1, Proposition 5.1 and (5.1) that12 F ′′ , ( q, V, θ ) = Z R n < − Lθ − qJ − i V F, θ > Gdx − Z R n ( q | J | + 12 | i V F | ) Gdx = Z R n < − aJ − i W F − L e θ, aJ + i W F + e θ > Gdx + Z R n < − qJ − i V F, aJ + i W F + e θ > Gdx − Z R n ( q | J | + 12 | i V F | ) Gdx = − ( a + q ) Z R n | J | Gdx − Z R n | i V + W F | Gdx + Z R n < − L e θ, e θ > Gdx Let q = − a , V = − W , one has the equivalence. (cid:3) TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 19 Entropy and entropy-stability We now introduce λ -entropy of connections on the trivial G -vector bundle E over R n .We shall show that along the Yang-Mills flow, the entropy is non-increasing. We also provethat the entropy of a homothetically shrinking soliton A ( x ) ∈ S x ,t is achieved exactly by F x ,t ( A ), provided that i V F = 0 for any non-zero vector V ∈ R n . Definition 6.1. Let A ( x ) be a connection on E . We define the entropy by (6.1) λ ( A ) = sup x ∈ R n ,t > F x ,t ( A ) . We first consider the invariance property of the entropy. Proposition 6.1. The entropy λ is invariant under translations and rescalings.Proof. Let A ( x ) be a connection on E . A translation of A ( x ) is a new connection, denotedby e A ( x ), of the form e A i ( x ) = A i ( x + x ) , where x is a point in R n . For any x ∈ R n and t > 0, we have F x − x ,t ( e A ) = F x ,t ( A ) . Hence λ ( e A ) = λ ( A ) . A rescaling of A ( x ) is a new connection, denoted by A c ( x ) of the form A ci ( x ) = c − A i ( c − x ) , where c is a positive number. Then, by setting y = c − x , we have F cx ,c t ( A c ) = ( c t ) Z R n | F c ( x ) | (4 πc t ) − n e − | x − cx | c t dx = ( c t ) Z R n c − | F ( c − x ) | (4 πc t ) − n e − | x − cx | c t dx = ( c t ) Z R n c − | F ( y ) | (4 πc t ) − n e − | cy − cx | c t c n dy = t Z R n | F ( y ) | (4 πt ) − n e − | y − x | t dy = F x ,t ( A ) . Hence λ ( A c ) = λ ( A ) . (cid:3) In the case that A ( x ) is a homothetically shrinking soliton, Proposition 6.1 explainswhy in Theorem 5.2 J and i V F do not violate the F-stability. Proposition 6.2. Let A ( x, t ) be a solution to the Yang-Mills flow on E . Then the entropy λ ( A ( x, t )) is non-increasing in t . Proof. Let t < t < T . Here T denotes the first singular time of the Yang-Mills flow. By(6.1), for any given ǫ > x , t ) such that(6.2) λ ( A ( x, t )) − ǫ ≤ F x ,t ( A ( x, t )) . Note that for any c > ≤ t < T , we have(6.3) F x ,c ( A ( x, t )) = Φ x ,c + t ( A ( x, t )) . By (6.3), the monotonicity formula (2.8), and the definition of entropy, we have F x ,t ( A ( x, t )) = Φ x ,t + t ( A ( x, t )) ≤ Φ x ,t + t ( A ( x, t )) = F x ,t + t − t ( A ( x, t )) ≤ λ ( A ( x, t )) . Together with (6.2), we see that λ ( A ( x, t )) ≤ λ ( A ( x, t )) . (cid:3) Definition 6.2. A homothetically shrinking soliton A ( x ) is called entropy-stable if it isa local minimum of the entropy, among all perturbations e A ( x ) such that || e A − A || C issufficiently small. In general the entropy λ ( A ) is not attained by any F x ,t ( A ). However if A ∈ S x ,t and i V F = 0 for any V ∈ R n , we will show that λ ( A ) is attained exactly by F x ,t ( A ). Wefirst examine the geometric meaning of i V F = 0 in the case that A ( x ) is a homotheticallyshrinking solition. Proposition 6.3. If A ( x ) is a homothetically shrinking soliton satisfying i V F = 0 forsome non-zero vector V , then A ( x ) is defined on a hyperplane perpendicular to V .Proof. Without loss of generality we assume A ( x ) is centered at (0 , 1) and let A ( x, t ) bethe homothetically shrinking Yang-Mills flow with A ( x, 0) = A ( x ). In the exponentialgauge, i.e. a gauge such that x j A j ( x ) = 0, we have for any t < λ > A j ( x, t ) = λA j ( λx, λ ( t − 1) + 1) = 1 √ − t A j ( x √ − t , 0) = 1 √ − t A j ( x √ − t )and(6.5) F ij ( x, t ) = 11 − t F ij ( x √ − t ) . Moreover the exponential gauge is uniform for all t < 1, i.e. x j A j ( x, t ) = 0.By assumption we have i V F ( x ) = 0. For simplicity let V = ∂∂x l . Then by (6.5) we have F jl ( x, t ) = 0 , ∀ j. Note that A ( x, t ) is a homothetically shrinking Yang-Mills flow, hence J l ( x, t ) = 12(1 − t ) x j F jl ( x, t ) = 0 . Then ∂∂t A l ( x, t ) = J l ( x, t ) = 0 . In particular, A l ( x, t ′ ) = A l ( x, t ) , ∀ t, t ′ < . TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 21 Then by (6.4), we have for any λ > t < A l ( x, t ) = λA l ( λx, λ ( t − 1) + 1) = λA l ( λx, t ) . Letting λ → 0, we see that(6.6) A l ( x, t ) = 0 . Note that 0 = F lj ( x, t ) = ∂ l A j − ∂ j A l + A l A j − A j A l = ∂ l A j ( x, t ) , so for any j , we have ∂ l A j ( x, t ) = 0and(6.7) A j ( x + cV, t ) = A j ( x, t ) , ∀ c ∈ R . For example if V = ∂∂x n , then by (6.6) and (6.7) we have A ( x , · · · , x n − , x n , t ) = A ( x , · · · , x n − , , t ) dx + · · · + A n − ( x , · · · , x n − , , t ) dx n − . In particular for V = ∂∂x n and in the exponential gauge, we have(6.8) A ( x , · · · , x n − , x n ) = A ( x , · · · , x n − , dx + · · · + A n − ( x , · · · , x n − , dx n − . This means that A ( x ) is defined on a hyperplane perpendicular to V , i.e. A ( x ) descendsto a trivial G -vector bundle over a hyperplane V ⊥ . (cid:3) The following Proposition is analogous to a corresponding result for self-shrinkers of themean curvature flow, see [8]. We follow closely the arguments given in [8]. Proposition 6.4. Let A ( x ) be a homothetically shrinking soliton centered at (0 , suchthat i V F = 0 for any non-zero V . Then the function ( x , t ) 7→ F x ,t ( A ) attains its strictmaximum at (0 , . In fact for any given ǫ > , there exists a constant δ > such that (6.9) sup {F x ,t ( A ) : | x | + | log t | ≥ ǫ } < λ ( A ) − δ. In particular, the entropy of A is achieved by F , ( A ) .Proof. We first show that (0 , 1) is a local maximum of the function ( x , t ) 7→ F x ,t ( A ).That is to show F ′ , ( q, V, 0) = 0 , ∀ q, V, and F ′′ , ( q, V, < , ∀ ( q, V ) = (0 , . In fact by the first variation formula (3.3) and Lemma 3.2 (a, b), we have dds | s =0 F x s ,t s ( A ) = F ′ , ( q, V, 0) = 0 . Let x s = sV, t s = 1 + sq . Note that J = 0, otherwise F would be vanishing, as showed inthe proof of Corollary 5.1, which violates the assumption that i V F = 0 for any non-zero V . Then by the second variation formula (4.4), we have for any ( q, V ) = (0 , 0) that12 F ′′ , ( q, V, 0) = − Z R n ( q | J | + 12 | i V F | ) Gdx < . For any fixed ( y, T ), where y ∈ R n and T > 0, we set x s = sy, t s = 1 + ( T − s . Note that ( x s , t s ) , s ∈ [0 , , 1) to ( y, T ). Let g ( s ) = F x s ,t s ( A ) . The remaining of the proof is to show that g ′ ( s ) ≤ s ∈ [0 , g ′ ( s ) = Z R n ˙ t s ( 4 − n t s + 14 | x − x s | ) | F | G s ( x ) dx + Z R n t s < ˙ x s , x − x s > | F | G s ( x ) dx. In the same way as in the proof of Lemma 3.2, for vector fields ϕ on R n we have Z R n ϕ p ( x − x s ) p | F | G s ( x ) dx = Z R n [2 t s ∂ p ( ϕ p ) | F | − t s ∂ i ϕ p F αpjβ F αijβ ] G s dx − Z R n t s ϕ p F αpjβ ( J αjβ − t s X αjβ ) G s dx, where X αjβ = ( x − x s ) p F αpjβ . Taking ϕ = ∂∂x p and noting that J j = x p F pj , we get Z R n ( x − x s ) p | F | G s ( x ) dx = − Z R n t s F αpjβ ( J αjβ − t s X αjβ ) G s dx, = − Z R n t s F αpjβ ( 12 x i − t s ( x − x s ) i ) F αijβ G s dx. Taking ϕ ( x ) = x − x s , we get Z R n | x − x s | | F | G s dx = Z R n [2( n − t s | F | ] G s dx − Z R n t s X αjβ ( J αjβ − t s X αjβ ) G s dx = Z R n [2( n − t s | F | + 2 | X | ] G s dx − Z R n t s X αjβ x i F αijβ G s dx. Hence we have g ′ ( s ) = − Z R n n − t s ˙ t s | F | G s ( x ) dx + 14 ˙ t s [ Z R n [2( n − t s | F | + 2 | X | ] G s dx − Z R n t s X αjβ x i F αijβ G s dx ] − t s y p Z R n t s F αpjβ ( x i − t s ( x − x s ) i ) F αijβ G s dx = 12 ˙ t s [ Z R n | X | G s dx − Z R n t s X αjβ x i F αijβ G s dx ] − t s y p Z R n t s F αpjβ ( x i − t s ( x − x s ) i ) F αijβ G s dx. TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 23 Set z = x − x s = x − sy . We have x = z + sy and X j = z i F ij . Then we get g ′ ( s ) = 12 ˙ t s [ Z R n (1 − t s ) | X | G s dx − Z R n t s X αjβ sy i F αijβ G s dx ] − t s y p Z R n t s F αpjβ ( z i + sy i − t s z i ) F αijβ G s dx = 12 ˙ t s [ Z R n (1 − t s ) | X | G s dx − Z R n t s X αjβ sy i F αijβ G s dx ] − t s y p Z R n ( t s − F αpjβ X αjβ G s dx − t s Z R n sy p F αpjβ y i F αijβ G s dx = 12 ˙ t s (1 − t s ) Z R n | X | G s dx − ( 12 s ˙ t s t s + t s ( t s − Z R n < X j , y i F ij > G s dx − st s Z R n | y i F ij | G s dx. For t s = 1 + ( T − s , we have g ′ ( s ) = − s [( T − s Z R n | X | G s dx + 2( T − st s Z R n < X j , y i F ij > G s dx + t s Z R n | y i F ij | G s dx ]= − s Z R n | ( T − sX j + t s y i F ij | G s dx ≤ . (cid:3) entropy-stability and F-stability In this section we shall show that the entropy-stability of a homothetically shrinkingsoliton such that i V F = 0 for any non-zero V implies F-stability. Theorem 7.1. Let A ( x ) be a homothetically shrinking soliton in S , such that i V F = 0 for any non-zero V . If A ( x ) is entropy-stable, then it is F-stable.Proof. We argue by contradiction. Assume that A ( x ) is F-unstable. By the definitionof F-stability there exists a 1-parameter family of connections A s ( x ) , s ∈ [ − ǫ, ǫ ], with θ s ( x ) := dds A s ( x ) ∈ C ∞ , such that for any deformation ( x s , t s ) of ( x = 0 , t = 1), we have(7.1) d ds | s =0 F x s ,t s ( A s ) < . We start from this to show that A is entropy-unstable. Let H : R n × R + × [ − ǫ, ǫ ] , H ( y, T, s ) = F y,T ( A s ) . In fact we will show that there exists ǫ > s with 0 < | s | ≤ ǫ ,(7.2) sup y,T H ( y, T, s ) < H (0 , , . Hence for s with 0 < | s | ≤ ǫ , λ ( A s ) < λ ( A ), which contradicts with our assumption. Step 1. We prove that there exists ǫ > s with 0 < | s | ≤ ǫ ,(7.3) sup { H ( y, T, s ) : | y | ≤ ǫ , | log T | ≤ ǫ } < H (0 , , . By the assumption that A ( x ) ∈ S , and Corollary 3.2, we have ∇ H (0 , , 0) = 0 . For any y ∈ R n , a ∈ R and b ∈ R , ( sy, as, bs ) is a curve through (0 , , b = 0, by (7.1) we have d Hds | s =0 ( sy, as, bs ) = d ds | s =0 F sy, as ( A bs )= b d ds | s =0 F sb y, a sb ( A s ) < . For b = 0 and ( a, y ) = (0 , d Hds | s =0 ( sy, as, 0) = d ds | s =0 F sy, as ( A )= − Z R n ( a | J | + 12 | i y F | ) Gdx< , where we used the assumption that i y F = 0 for y = 0 and its implication that J = 0.Hence the Hessian of H at (0 , , 0) is negative definite and H has a local strict maximumat (0 , , ǫ ∈ (0 , ǫ ] such that if 0 < | y | + | log T | + | s | ≤ ǫ , then H ( y, T, s ) < H (0 , , . In particular for any s with 0 < | s | ≤ ǫ , we havesup { H ( y, T, s ) : | y | ≤ ǫ , | log T | ≤ ǫ } < H (0 , , . Step 2. We prove that there exists R > T,s H ( y, T, s ) < H (0 , , , for | y | ≥ R . Denote the support of θ s by Ω s and Ω = S s ∈ [ − ǫ,ǫ ] Ω s . Then on R n \ Ω, F s = F . Hence H ( y, T, s ) = T Z Ω | F s | (4 πT ) − n e − | x − y | T dx + T Z R n \ Ω | F | (4 πT ) − n e − | x − y | T dx ≤ T Z Ω | F s | (4 πT ) − n e − | x − y | T dx + H ( y, T, . Note that for | y | ≥ 1, there exists δ > H ( y, T, ≤ H (0 , , − δ , see Propo-sition 6.4. Let M = sup {| F s ( x ) | : s ∈ [ − ǫ, ǫ ] , x ∈ R n } , D = sup x ∈ Ω | x | and | Ω | = R Ω dx .Then for | y | ≥ D + R with R ≥ 1, we have H ( y, T, s ) ≤ M | Ω | T (4 πT ) − n e − R T + H (0 , , − δ. Let f ( r ) = r − n − e − r , r > 0, which is uniformly bounded. Note that n ≥ 5. Thenas R → ∞ , T − n e − R T = f ( TR ) R − n → 0, uniformly in T > 0. Hence we can choosesufficiently large R such that for | y | ≥ D + R := R , we have H ( y, T, s ) ≤ H (0 , , − δ . TABILITIES OF HOMOTHETICALLY SHRINKING YANG-MILLS SOLITONS 25 Step 3. We prove that exists T > y,s H ( y, T, s ) < H (0 , , , for | log T | ≥ T . We first consider the case that T is large. Note that for any T > H ( y, T, s ) = T Z Ω | F s | (4 πT ) − n e − | x − y | T dx + T Z R n \ Ω | F | (4 πT ) − n e − | x − y | T dx ≤ T Z Ω | F s | (4 πT ) − n e − | x − y | T dx + H ( y, T, ≤ M | Ω | T (4 πT ) − n + H ( y, T, . By Proposition 6.4, there exists δ > H ( y, T, ≤ H (0 , , − δ when T ≥ T ≥ H ( y, T, s ) ≤ H (0 , , − δ , for T ≥ T . Note that M = sup {| F s ( x ) | : s ∈ [ − ǫ, ǫ ] , x ∈ R n } . Hence for any T > 0, we have H ( y, T, s ) = F y,T ( A s ) = T Z R n | F s | (4 πT ) − n e − | x − y | T dx ≤ M T . Thus there exists T > y,s H ( y, T, s ) < H (0 , , , for T ≤ T . Combing (7.6) and (7.7), we get (7.5).Step 4. Set U = { ( y, T ) : | y | ≤ R , | log T | ≤ T } \ { ( y, T ) : | y | < ǫ , | log T | < ǫ } . We now prove that there exists ǫ ≤ ǫ such that for any s with | s | ≤ ǫ ,(7.8) sup { H ( y, T, s ) : ( y, T ) ∈ U } < H (0 , , . Note that U is a compact set which does not contain (0 , δ > U H ( y, T, ≤ H (0 , , − δ. 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Albert-Ludwigs-Universit¨at Freiburg, Mathematisches Institut, Eckerstr. 1, 79104 Freiburg,Germany; New Address: Institute of Mathematics, Academy of Mathematics and SystemsScience, Chinese Academy of Sciences, Beijing 100190, China E-mail address : [email protected] School of Mathematical Sciences and Wu Wen-Tsun Key Laboratory of Mathematics,University of Science and Technology of China, Hefei 230026, Anhui Province, China E-mail address ::