The Gradient Flow for Gauged Harmonic Map in Dimension Two I

The Gradient Flow for Gauged Harmonic Map in Dimension Two I

Yong YU

Department of Mathematics, University of Iowa

ABSTRACT: In this article, we study a gradient flow associated with a gauged harmonic map energy in dimension two. Some specific properties are considered, for instances bubbling analysis, asymptotic behavior and removability of singularities.

Mathematics Subject Classification (2010): Primary 53C05, 58E15 · Secondary 35K40, 35K55

TABLE OF CONTENTS

I. Introduction 2

I.1. Gauged Harmonic Map and Gradient Flow . . . 2

I.2. Main Results . . . 3

II. Local Existence 5 II.1. Gauge Equivalent Flow and Its Extension . . . 5

II.2. Estimates for linear heat equation on vector bundles . . . 6

II.3. Local Existence for the Gradient Flow . . . 8

III. Energy Inequalities and Criterion for First Singular Time 12 III.1. Local Energy Inequalities . . . 12

III.2. Bochner-type Inequality . . . 13

III.3. -Regularity . . . 15

III.4. Criterion for First Singular Time . . . 18

IV. Bubbling Analysis 22 IV.1. Convergence of the gradient flow . . . 22

IV.2. Bubbling Analysis . . . 25

V. Asymptotic Behavior 32 VI. Removability of Singularites 35 References . . . 40

I.

The theories of harmonic map and Yang-Mills play fundamental roles in the study of both physics and geometry. In this work, we couple these two theories and study a model of gauged harmonic map. Our motivations stem from the work of [1], [2], [11], [16] and the references therein. But as one will see, we define our model in more general settings, which involve a non-Abelian structure group G and a fibre bundle E , whose typical fibreN is a closed G -invariant Riemannian manifold.

I.1.

LetM and N be two closed Riemannian manifolds of dimensions m and n, respectively. We assume that M is equipped with a Riemannian metric g and N is isometrically embedded into R^L. Suppose that

G ⊆ SO(L)

is a compact Lie group with Lie algebra g. Naturally, if we identify an element inN as a column vector in R^Lthrough the isometrical embedding, thenG induces a smooth left action on N by the left multiplication of matrix. In this article, we require thatN is G -invariant. That is

g ·N ⊆ N , ∀ g ∈G .

In order to introduce the model of gauged harmonic map, we need some geometric terminology. Let {Uα} be a finite open covering ofM . On Uα∩Uβ, we define a smooth map

gα,β :Uα∩Uβ7−→G .

Obviously, if {gα,β} satisfies the co-cycle condition, then it determines a principalG -bundle, denoted by P, over M . By P, we can construct a fibre bundle E = P ×_G N via the left action of G on N which was discussed above. It is clear thatE is a sub-bundle of F = P ×_G R^L.

The variables in the theory of gauged harmonic map are a connection 1-form A onP and a section φ of E . Locally on Uα, A and φ can be represented as

Aα:Uα7−→ g and φα:Uα7−→N , respectively. Moreover, onUα∩Uβ,

A_α= Ad_gα,β(A_β) − d g_α,β· gβ,α and φ_α= g_α,βφ_β,

where Ad is the adjoint representation ofG . The energy functional associated with (A, φ) is defined by E(A, φ) =

Z

M

e(A, φ) dvg, (1.1)

where

e(A, φ) = 1

2 |FA|²+ |DAφ|²

is the energy density. Note that in the definition of e(A, φ), F_A is the curvature 2-form, while D_Aφ is the covariant derivative induced by A. The norms in e(A, φ) are defined in a natural way, using the metrics on M , R^L and the Killing form of the Lie algebra g. With the energy functional E(A, φ), we define gauged harmonic map to be a critical point of (1.1). More precisely,

Definition 1.1 (Gauged Harmonic Map). A section φ ∈ Ω⁰(E ) is called gauged harmonic map if there is a connection A such that (A, φ) satisfies the following Euler-Lagrange equation of (1.1):







D^∗_AF_A= − (g_lφ, D_Aφ) g_l;

D^∗_AD_Aφ = (D_Aν_i(φ), D_Aφ) ν_i(φ).

(1.2)

D_A^∗ is the formal adjoint operator of DA. {gl} (l = 1, ..., k) is an orthonormal basis of the Lie algebra g under the inner product h·, ·i which is induced from the Killing form on so(L).

{νi(φ)} , i = 1, ..., L − n

is an orthonormal frame of the normal bundle ( TN )^⊥ at φ. Similarly as in the work of harmonic map and Yang-Mills, we can introduce a gradient flow associated with the energy functional (1.1) as follows:







∂tA = −D^∗_AFA− (glφ, DAφ) gl;

∂_tφ = −D_A^∗D_Aφ + (D_Aν_i(φ), D_Aφ) ν_i(φ).

(1.3)

Note that (1.2) is gauge invariant under the gauge transformation s · (A, φ) = (s · A, s · φ) ,

where s · A = Ads(A) − ds · s⁻¹, s · φ = s φ, while (1.3) is also gauge invariant under a time-independent gauge transformation s.

I.2.

There are two directions in the study of gauged harmonic map. The first one is to reduce the problem to a first-order Bogomol’nyi type equation (vortex equation) by studying the lowest bound of energy functional in a homotopy class. For instances, [1], [2], [11], [16] and the references therein. To solve the vortex equation, particularly in Abelian case, one can apply either the method of Taubes (see [11]) or a stability criterion based on Hitchin-Kobayashi correspondence (see [2]). In fact, Taubes’ method works pretty well when the base manifoldM is a Riemannian surface or C, while the stability method has its application in the case whenM is a K¨ahler manifold with higher complex dimension. Our approach follows the second direction.

That is to study the gradient flow associated with the energy functional (1.1). Along this direction, many works have been carried out in the theory of harmonic map (see [10], [18], [20]), Yang-Mills (see [13], [15], [17], [22]) and Yang-Mills-Higgs (see [5]-[7]). As is well-known, the critical dimension for the heat flow of harmonic map is 2, while the critical dimension for the Yang-Mills flow is 4. Therefore, when dim(M ) = 2, our model is subcritical for the Yang-Mills fields and critical for the section ofE .

We now describe the organization of this article. In Section II, we prove the local existence of the gradient flow (1.3) with smooth initial data (A0, φ0). More precisely, we show that

Theorem 1.2. There is a T > 0 so that the gradient flow (1.3) admits a smooth solution on [ 0, T ) with the given smooth initial data (A0, φ0). For p > dim(M ), T can be shown to depend on the W^2,p-norm of (A0, φ0).

Theorem 1.2 works for any dimension. Start from Section III. We assume that dim(M ) = 2 and study some specific properties associated with the gradient flow (1.3), for instances bubbling analysis, asymptotic behav- ior and removability of singularities. Section III is a preparation, in which we show local energy inequalities, Bochner-type inequality and -regularity. A criterion is given in Section III.4 for the first singular time T0

of the gradient flow (1.3). If T0< ∞, then the bubbling phenomenon occurs at T0. In this case, we show that

Theorem 1.3. Suppose that dim(M ) = 2. If the first singular time T0 is finite, then there exist a set of finitely many points in M , denoted by {xi}, so that for all k ∈ N,

(A(t), φ(t)) −→ (A(T0), φ(T0)), in C_loc^k (M \ {xi}), as t ↑ T0.

Moreover, there exist finitely many non-trivial harmonic maps from R² into N , denoted by {φ^∗s}, so that the following energy identity holds:

lim

t↑T₀

Z

M

e(t) dv_g= Z

M

e(T₀) dv_g+1 2

X

s

Z

R²

|∇φ^∗_s|² dx. (1.4)

Conventionally, {φ^∗_s} in Theorem 1.3 are called bubbles. From (1.4), one realizes that due to the existence of singluar points, the gradient flow (1.3) loses some energy at T₀. Moreover, the lost energy can be recovered by finitely many harmonic maps from R²intoN . Different from the assumptions in Theorem 1.3, in Section V, we suppose that the gradient flow (1.3) admits a global smooth solution on [ 0, ∞). We are interested in the asymptotic behavior of the global solution as t ↑ ∞. In fact, we have

Theorem 1.4. Suppose that (A, φ) is a global smooth solution of (1.3). Then there exist tk ↑ ∞ and a finite covering {B_i^∗} of M so that the followings hold:

(1). For each k and i, (A(tk), φ(tk)) is gauge equivalent to some smooth (A^∗_k,i, φ^∗_k,i) on B_i^∗. Define A^∗_k|_B∗

i

= A^∗_k,i, φ^∗_k|_B∗ i

= φ^∗_k,i, for all i.

Then there exists a principalG -bundle Pk overM so that A^∗_k is a smooth connection on Pk and φ^∗_k is a smooth section on the associated fibre bundle Ek =Pk×_GN ;

(2). As k → ∞, we have a smooth principal G -bundle P^∗ overM so that Pk−→P^∗, Ek −→E^∗=P^∗×_G N .

Here, the convergence of principal G -bundles and fibre bundles are defined to be the C^∞-convergence of the associated transition functions;

(3). There are a W^1,2-connection A∞ on P^∗ and a W^1,2-section φ∞ on E^∗ so that A∞ and φ∞ are smooth away from points in Σ, where Σ is a finite subset of M . Moreover,

(A^∗_k, φ^∗_k) −→ (A_∞, φ_∞), in C_loc^∞(M \ Σ), as k → ∞;

(4). (A_∞, φ_∞) solves (1.2) smoothly away from the points in Σ.

Similarly as in the case of harmonic maps (see [14]), we can remove the singularities in Σ from (A_∞, φ_∞) and show in Section VI that

Theorem 1.5 (Removability of Singularities). (A_∞, φ_∞) is a global smooth solution of (1.2) onM .

II.

Let A0 be a smooth connection 1-form on P and φ0 ∈ Ω⁰(E ) be a smooth section. Here in the following, we study the local existence of the gradient flow (1.3) with the given initial data (A₀, φ₀). One should refer to [21] for some standard terminology in the gauge field theory. Sobolev spaces of sections of vector bundles are also introduced in [21].

Now we sketch the plan of this section. Note that the first equation in (1.3) is just partially parabolic.

Therefore, in Section II.1 below, we use gauge transformation, similarly as the work of Donaldson, to reduce the gradient flow (1.3) into a parabolic gauge equivalent flow. Furthermore, we use the projection near the manifoldN ,→ R^Lto get rid of the constraint on the range of unknown section. In such a way, we obtain an extended gauge equivalent flow. The linear theory of parabolic equation on vector bundles and contraction mapping theorem then can be applied to attain a unique smooth solution of the extended gauge equivalent flow. We then show, similarly as the heat flow of harmonic map (see [10]) and liquid crystal flow (see [9] and [20]), that if the initial section lies in Ω⁰(E ), then the solution of the extended gauge equivalent flow must be a solution of the parabolic gauge equivalent flow, which, furthermore, implies a solution of the original gradient flow (1.3).

II.1.

We reduce the gradient flow (1.3) into a parabolic gauge equivalent flow. Suppose that S is a gauge trans- formation and A, ψ
is gauge equivalent to (A, φ) via S¯ ⁻¹. That is

A = S¯ ⁻¹· A, ψ = S⁻¹· φ.

It is clear that if (A, φ) satisfies (1.3), then A, ψ
solves¯







∂tA = −D¯ ^∗_A¯FA¯− (glψ, DA¯ψ) gl+ DA¯ S⁻¹· ∂tS
;

∂tψ = −D^∗_A¯DA¯ψ + (DA¯νi(ψ), DA¯ψ) νi(ψ) − S⁻¹· ∂tS
· ψ.

(2.1)

Let Aref be a smooth reference connection and suppose that ¯A = Aref+ a. By requiring that

S⁻¹· ∂tS = −D_ref^∗ a, (2.2)

one may then rewrite the equation (2.1) in terms of (a, ψ) as follows:







∂ta + ∆refa = f (a, ψ) − (glψ, Drefψ)gl− D_ref^∗ Fref;

∂tψ + ∇^∗_ref∇refψ = (DA¯νi(ψ), DA¯ψ) νi(ψ) + 2a^k∇ref,kψ + a^kakψ,

(2.3)

where Fref is the curvature 2-form of Aref, ∇ref is the induced covariant derivative and f (a, ψ) = a × Fref+ a × ∇refa − (glψ, aψ)gl+ a × a + a × a × a.

In the above, × denotes any multi-linear map with smooth coefficients.

∆_ref = D^∗_refD_ref+ D_refD^∗_ref

is the Hodge Laplacian. System (2.3) is called parabolic gauge equivalent flow corresponding to (1.3).

Note that the unknown variable ψ in (2.3) can be represented locally as a map intoN . To get rid of the constraint on the range of ψ, we need a smooth projection

Π :N3δ 7−→N , for some δ > 0.

HereN3δ is the 3 δ-neighborhood ofN in R^L. Let ρ1 be a smooth non-negative function such that

ρ1(s) =















1, if s ∈ [0, δ];

≤ 1, if s ∈ [δ, 2 δ];

0, if s ≥ 2 δ.

ρ2is a cut-off function, which is defined by

ρ2(x) = ρ1(dist (x,N )) , ∀ x ∈ R^L. Obviously, ρ₂ isG -invariant. That is ∀ x ∈ R^L, g ∈G , we have

ρ2(g · x) = ρ2(x).

By the projection Π and the cut-off function ρ₂, one can define an extended gauge equivalent flow as follows:







∂ta + ∆refa = f (a, ψ) − (glψ, Drefψ)gl− D^∗_refFref;

∂tψ + ∇^∗_ref∇refψ = ρ2(ψ) (DA¯νi(ψ), DA¯(Πψ)) νi(ψ) + 2a^k∇ref,kΠψ + a^kakΠψ
.

(2.4)

Here ψ is an unkown section onF . νi(ψ) should be understood as the i-th normal direction at Πψ.

II.2.

Denote by Q_T the cylinderM × [0, T ]. With the given f ∈ L^p(Q_T) and φ⁰ ∈ W^2,p(M ; F ), we study the linear parabolic system defined as follows:







∂tφ + ∇^∗_ref∇refφ = f ; φ|_t=0= φ⁰.

(2.5)

Basically, there are two estimates important to us. The first one is the W_p^2,1 - estimate for the solution φ of (2.5). Another one is the L^∞W^1,p-estimate. We consider these two estimates in Proposition 2.1 and 2.2, respectively. In the following, p > 2 is a fixed constant.

Proposition 2.1. The system (2.5) admits a unique solution such that kφk_W^2,1

p (Q_T). kf kL^p(Q_T)+ φ⁰

_W2,p.

Proof. Firstly, we reduce the system (2.5) into the case in which φ⁰≡ 0. Let Σ = {Uα} be the finite open covering ofM , by which the principal bundle P is defined. Suppose that

ρα∈ C_c^∞(Uα)

is a sequence of non-negative functions subordinate to the covering Σ. We require that X

α

ρ²_α≡ 1, in M .

Fix an α. ραφ⁰_α is in fact a map from R^m to R^L. Here φ⁰_αis the local representation of φ⁰ inUα. Define ψ_α= Γ_t∗ ραφ⁰_α
,

where Γt(x) = Γ(x, t) is the standard heat kernel on R^m and ∗ is the spatial convolution operator on R^m. One can easily check that

ess sup

t>0

kψαk^p_W2,p . φ⁰_α

p

W^2,p(Uα).

Denote by ¯ψ_αthe restriction of ψ_α onUαand patch them together by setting Ψ_α(·, t) =X

γ

ρ_γg_α,γψ¯_γ(·, t), in Uα.

We claim that

1. OnUα∩Uβ, Ψ_α= g_α,βΨ_β. Hence, Ψ(·, t) ∈ W^2,p(M ; F ) by our construction;

2. At t = 0,

Ψα(·, 0) =X

γ

ργgα,γψ¯γ(·, 0) =X

γ

ρ²_γgα,γφ⁰_γ =X

γ

ρ²_γφ⁰_α= φ⁰_α.

Therefore, Ψ(·, 0) = φ⁰;

3. Ψ admits a W_p^2,1-estimate shown as follows:

ess sup

t>0

Z

M

|∂_tΨ|^p dv_g+ kΨk^p_W2,p

 .

φ⁰

p

W^2,p. (2.6)

The estimate on ∂tΨ in (2.6) can be derived by noticing that

∂tψα= ∆ψα, in R^m. Here ∆ is the standard Laplace operator in R^m.

We define Φ = φ − Ψ. It is clear that Φ satisfies







∂_tΦ + ∆_refΦ = ˆf := f − (∂_tΨ + ∆_refΨ) ; Φ|_t=0= 0.

(2.7)

Notice (2.6), we know that ˆf ∈ L^p(QT). Now we apply Proposition 2.7 in [22] and imply that kΦk_W2,1

p (Q_T). k ˆf k_Lp(Q_T). kf kL^p(Q_T)+ φ⁰

_W2,p. (2.8)

Combine the above inequality with (2.6), we have kφk_W2,1

p (QT). kf kL^p(Q_T)+ φ⁰

_W2,p. The proof is then finished.

As for the L^∞W^1,p-estimate for the solution of (2.5), we have

Proposition 2.2. Let φ be the unique solution of (2.5). Then ess sup

t∈[0,T ]

kφk^p_W1,p . kf k^pL^p(Q_T)+ φ⁰

p W^2,p.

Proof. Act ∇ref on both sides of (2.7) and inner product with p |∇refΦ|^p−2∇refΦ. One has d

dt Z

M

|∇refΦ|^p+ p Z

M

|∇refΦ|^p−2∇refΦ, ∇ref∆refΦ

= p Z

M

|∇refΦ|^p−2∇refΦ, ∇reffˆ .

In the above integral and the integral in the following, we omit dvg for convenience. Integrate by parts for the right-hand side and the second term on the left-hand side above. Therefore,

d dt

Z

M

|∇refΦ|^p+ p Z

M

|∇refΦ|^p−2|∆refΦ|²− p Z

M

|∇refΦ|^p−2

∆refΦ, ˆf

=

= p (p − 2) Z

M

|∇refΦ|^p−4

∆refΦ − ˆf , ∇²_refΦ (∇refΦ, ∇refΦ) + FrefΦ (∇refΦ, ∇refΦ) .

Fix an arbitrary τ ∈ [0, T ] and integrate the above equality with respect to t from 0 to τ . One may imply by (2.8) and H¨older’s inequality that

ess sup

τ ∈[0,T ]

Z

M

|∇refΦ|^p. kf k^pL^p(QT)+ φ⁰

p W^2,p. Notice (2.6). The proof is then finished.

Simlar arguments can be applied to 1-forms. In fact, we have

Proposition 2.3. Suppose that f ∈ L^p([0, T ]; L^p(T^∗M ⊗ AdP)) and a⁰∈ W^2,p(T^∗M ⊗ AdP). If a is the unique solution of the system:







∂_ta + ∆_refa = f ; a|_t=0= a⁰,

(2.9)

where ∆_ref is the Hodge Laplacian, then one has kak_W2,1

p (Q_T)+ ess sup

t∈[0,T ]

kak_W1,p . kf kL^p(QT)+ a⁰

_W2,p.

II.3.

In this section, we assume that p > m.

a⁰∈ Ω¹(AdP) and ψ⁰∈ Ω⁰(F )

are initial datum corresponding to the extended gauge equivalent flow (2.4). Without loss of generality, we choose T < 1 and suppose that V_p,T^g and V_p,T^s are closures of

C₀^∞+ [0, T ]; Ω¹(AdP)
and C₀^∞+ [0, T ]; Ω⁰(F )

under the norms

k · k^p_Vg p,T

= ess sup

t∈[0,T ]

k · k^p_W1,p(T∗M ⊗AdP)+ Z T

0

Z

M

∇²_ref · 

p

and

k · k^p_Vs

p,T = ess sup

t∈[0,T ]

k · k^p_W1,p(M ;F)+ Z T

0

Z

M

∇²_ref · 

p,

respectively. Here, all smooth 1-forms in C₀^∞₊ [0, T ]; Ω¹(AdP)
and sections in C₀^∞+ [0, T ]; Ω⁰(F )
take zero initial values. By V_p,T^g and V_p,T^s , we define V_p,T := V_p,T^g × V_p,T^s , which is equipped with the norm

k · k_Vp,T = k · k_V^g

p,T + k · k_V^s

p,T. Notice that, by Sobolev embedding,

kf k_L^∞_(QT). kf k^Vp,T, ∀ f ∈ Vp,T. (2.10)

Proposition 2.4. With the given smooth initial datum a⁰, ψ⁰
, there exists a T > 0 such that the extended gauge equivalent flow (2.4) admits a unique smooth solution in [ 0, T ). T depends on the p and W^2,p-norm of (a⁰, ψ⁰).

Proof. Let f ≡ 0. We solve the homogeneous equation of (2.9) and (2.5) with the given initial datum a⁰and ψ⁰. The solutions are denoted by a¹ and ψ¹, respectively. By Proposition 2.1-2.3, we have

a¹, ψ¹
_V

p,T .

a⁰, ψ⁰

_W2,p. (2.11)

Decompose the unknow variable (a, ψ) as a = a¹+ ¯a, ψ = ψ¹+ ¯ψ. Therefore, one can rewrite the equation (2.4) in terms of ¯a, ¯ψ
as follows:







∂t¯a + ∆refa = f¯ 1 ¯a, ¯ψ
;

∂tψ + ∇¯ ^∗_ref∇refψ = g¯ 1 ¯a, ¯ψ
,

(2.12)

where f₁ ¯a, ¯ψ
and g1 a, ¯¯ ψ
are defined to be

f ¯a + a¹, ¯ψ + ψ¹
− gl ψ + ψ¯ ¹
, Drefψ + D¯ refψ¹
gl− D^∗_refFref

and g ¯a + a¹, ¯ψ + ψ¹
, respectively. Here f (a, ψ) is defined in (2.3) and g(a, ψ) stands for the right-hand side of the second equation in (2.4).

We use the contraction mapping theorem to solve (2.12) with 0 initial datum. In the following, C is a suitably large constant depending on p,M , N , Drefand the W^2,p-norms of a⁰and ψ⁰. Fix ¯a_∗, ¯ψ_∗
∈ Br₀,T, where r0< 1 and Br₀,T is the ball in Vp,T with center 0 and radius r0. We consider the system







∂t¯a + ∆ref¯a = f1 ¯a_∗, ¯ψ_∗
;

∂tψ + ∇¯ ^∗_ref∇refψ = g¯ 1 a¯_∗, ¯ψ_∗
,

(2.13)

with 0 initial value. By (2.10)-(2.11), we know that

f₁ ¯a_∗, ¯ψ_∗
∈ L^p([0, T ]; L^p(T^∗M ⊗ AdP)) and can be estimated by

Z

Q_T

f1 a¯_∗, ¯ψ_∗
 

p≤ C T.

Apply Proposition 2.3, there exists a unique solution ¯a of the first equation in (2.13) and moreover, k¯ak^p_Vg

p,T

≤ C T. (2.14)

Similar arguments can be applied to ¯ψ which is the solution for the second equation in (2.13). Note that Z

Q_T

g1(¯a∗, ¯ψ∗) 

p≤ C

 T +

Z

Q_T

∇refψ¯∗ 

2p+ ∇refψ¹

2p

. (2.15)

We estimateR

QT

∇refψ¯_∗ 

2p in (2.15). In one way, it can be bounded by Z T

0

∇refψ¯_∗

p L^∞(M )

Z

M

∇refψ¯_∗ 

p≤ ess sup

t∈[0,T ]

Z

M

∇refψ¯_∗ 

p· Z T

0

∇refψ¯_∗

p L^∞(M ). In another way, by Sobolev embedding, one may estimate the last term above by

ess sup

t∈[0,T ]

Z

M

∇refψ¯_∗ 

p· Z T

0

∇refψ¯_∗

p

W^1,p≤ C ¯ψ_∗

2p V^s_p,T. Since ¯ψ∗ ∈ Br0,T, we know that

Z

Q_T

∇refψ¯∗ 

2p≤ C ¯ψ∗

2p

V_p,T^s ≤ C r₀^2p. Similarly, forR

Q_T|∇refψ¹|^2p, we have Z

Q_T

∇_refψ¹ 

2p≤ C ess sup

0≤t≤T

Z

M

∇_refψ¹ 

p· Z

Q_T

∇_refψ¹ 

p+ ∇²_refψ¹

p≤ C

 T +

Z

Q_T

∇²_refψ¹ 

p .

Therefore, one can estimate g₁(¯a_∗, ¯ψ_∗) as follows:

Z

QT

g1 ¯a_∗, ¯ψ_∗
 

p ≤ C



T + r^2p₀ + Z

QT

∇²_refψ¹ 

p

. (2.16)

Then by Proposition 2.1 and 2.2, we have

¯ψ

p

V^s_p,T ≤ C



T + r^2p₀ + Z

Q_T

∇²_refψ¹ 

p

. (2.17)

By (2.14) and (2.17), we know that if T and r₀are suitably small, the solution ¯a, ¯ψ
of (2.13) lies in Br0,T. Here we used the absolute continuity of R

Q_T

∇²_refψ¹ 

p. Now we can construct a nonlinear operator which sends (¯a_∗, ¯ψ_∗) ∈ Br₀,T to the unique solution of (2.13). Clearly, this nonlinear operator is also a contraction mapping between Br₀,T and itself when r0 and T are suitably small. The local existence for the extended gauge equivalent flow is then obtained. The smoothness of the solution can be easily obtained by standard parabolic estimates. We omit the arguments here.

In the following, we show that the smooth solution for the extended gauge equivalent flow (2.4) is also a solution for the parabolic gauge equivalent flow (2.3) if the initial section is a section of the fibre bundleE . In fact, we have

Proposition 2.5. With given initial data (a⁰, φ⁰) ∈ Ω¹(AdP) × Ω⁰(E ), there exists a T > 0 such that the parabolic gauge equivalent flow (2.3) admits a unique smooth solution in [ 0, T ), where T depends on p and the W^2,p-norm of (a⁰, φ⁰).

Proof. Suppose that (a, ψ) is the unique solution of (2.4) with the given initial datum (a⁰, φ⁰). By the regularity of the extended gauge equivalent flow (2.4), we know that when T is small enough, ψ takes its value in the δ-neighborhood of N . Therefore, ρ2(ψ) ≡ 1 in [ 0, T ) and the second equation in (2.4) can be read as

∂tψ + ∇^∗_ref∇refψ = (DA¯νi(ψ), DA¯Πψ) νi(ψ) + 2a^k∇ref,kΠψ + a^kakΠψ. (2.18) Here ¯A = Aref+ a. Define ρ = ¹₂|ψ − Πψ|². Then by Lemma 2.6 below, we know that

∂_tρ + ∇^∗∇ρ ≤ 0.

Here −∇^∗∇ = ∆_M is the Laplace-Beltrami operator on the mainifoldM . The standard maximum principle implies that ρ ≡ 0 in [ 0, T ). The proof is then finished.

We complete the proof of Proposition 2.5 by showing Lemma 2.6 in the following.

Lemma 2.6. Let ρ be as in the proof of Proposition 2.5. Suppose that on [ 0, T ), (2.18) holds. Then

∂tρ + ∇^∗∇ρ = − |∇refψ − ∇ref(Πψ)|², ∀ t ∈ (0, T ).

Proof. By standard calculations, we know that

∂tρ + ∇^∗∇ρ = − |∇refψ − ∇ref(Πψ)|²+ (2.19)

+(ψ − Πψ) ·

 1

√g

∂

∂xj

√

g A^j_refΠψ

− (ψ − Πψ) · ∇^∗∇ψ+

+(ψ − Πψ) · A^k_ref∂kΠψ
+ (ψ − Πψ) · A^k_refAref,kΠψ
+ (ψ − Πψ) · (∂tψ + ∇^∗_ref∇refψ) .

We label from (I) to (VI) the six terms on the right-hand side of (2.19). Now we expand the right-hand side of (2.18) as follows so that we can plug it into the (VI)-th term in (2.19).

∂tψ + ∇^∗_ref∇refψ = (νi, ∇^∗∇Πψ) νi− νi, A^k_ref∂kΠψ
νi− νi, a^k∂kΠψ
νi− (2.20)

− νi, A^k_refAref,kΠψ
νi−

 νi, 1

√g∂j

√

g A^j_refΠψ

νi− νi, a^kAref,kΠψ
νi−

− νi, a^j∂jΠψ
νi− νi, A^k_refakΠψ
νi− νi, a^kakΠψ
νi+ 2 a^k∇ref,kΠψ + a^kakΠψ.

We label from (1)-(11) the terms on the right-hand side of (2.20). Notice that (ψ − Πψ) ⊥ TNψ.

Therefore, we can cancel some terms in (2.19) after we plug (2.20) into the (VI)-th term on the right-hand side of (2.19). In fact, (II) and (5), (III) and (1), (IV) and (2), (V) and (4) are the pairs, which can be cancelled out. In (2.20) itself, we see that (3) + (6) + (7) + (8) + (10) give us a tangent vector at ψ. It is orthogonal to ψ − Πψ. Obviously, (9) + (11) is also a tangent vector at ψ. Therefore, only the (I)-th term on the right-hand side of (2.19) remains after cancellation. The proof is then finished.

As a corallary of Proposition 2.5, we can show that the gradient flow (1.3) admits a local regular solution with the initial data (A0, φ0) given at the beginning of Section II. In fact, set (a⁰, φ⁰) = (A0− Aref, φ0). We can find a smooth solution (a, ψ) of the parabolic gauge equivalent flow by Proposition 2.5. In the rest, one just needs to solve the equation in (2.2) with the initial condition:

S(0) = Id.

Obviously,

A = S · (Aref+ a) , φ = S · ψ

provides us with a solution of (1.3). Moreover, we have the following energy identity,

Proposition 2.7. If (A, φ) is a regular solution of the gradient flow (1.3) in [ 0, T ), then d

dt Z

M

e(A, φ) dvg+ Z

M

|∂tA|²+ |∂tφ|²dvg= 0, ∀ t ∈ (0, T ).

The proof for this proposition is simple. We inner product ∂tA and ∂tφ on both sides of the first and second equations in (1.3), respectively. Here one may use the fact that ∂tφ is orthogonal to the normal vectors νi(φ). Then integrating by parts, the proof can be achieved.

III.

The main purpose of this section is to study a criterion for the first singular time of the gradient flow (1.3). Before that, we consider local energy inequalities, Bochner-type inequality and -regularity in Section III.1-3, respectively. The criterion will be given in Section III.4. In this section, all balls B_R(x₀) are geodesic balls with R < i(M ), where i(M ) is the infimum of the injectivity radius of each point x ∈ M .

III.1.

In this section, we prove the following local energy inequalities for a solution of the gradient flow (1.3).

Proposition 3.1. Suppose that (A, φ) is a smooth solution of (1.3) in M × [0, T0). Then for all x0∈M , 0 < R < i(M ) and 0 ≤ S < T < T0, we have

Z

B_R/2(x₀)

e(A, φ) dv_{g     T}

≤ Z

B_R(x₀)

e(A, φ) dv_{g     S}

+ C E(S) R⁻²(T − S) (3.1) and

(3.2) Z

B_R/2(x₀)

e(A, φ) dvg

    _S

≤ Z

B_R(x₀)

e(A, φ) dvg

    _T

+ C E(S) R⁻²(T − S) + C Z T

S

Z

M

| ∂tA |²+ | ∂tφ |², where E(S) is the total energy of (A, φ) at time S and C is independent of x0, (A, φ), R, S and T .

Proof. Choose x0and R as in the assumption of Proposition 3.1. Define f a cut-off function such that f ≡ 1 in BR/2(x0), f ≡ 0 outside BR(x0) and |f | ≤ 1 onM . Moreover, we assume that |∇f| ≤ C/R, where C > 0 is an universal constant.

Inner product f²∂_tφ on both sides of the second equation in (1.3) and integrate by parts. We imply that 1

2 d dt

Z

M

f²|DAφ|²dvg+ Z

M

f²|∂tφ|²dvg+ 2 Z

M

f (DAφ, ∂tφ df ) dvg= Z

M

DAφ, f²∂tA · φ
dvg. Inner product f²∂_tA on both sides of the first equation in (1.3) and integrate by parts. One has

1 2

d dt

Z

M

f²|FA|²dvg+ Z

M

f²|∂tA|²dvg+ 2 Z

M

f hFA, df ∧ ∂tAi dvg= − Z

M

DAφ, f²∂tA · φ
dvg. Sum the above two equalities. One can show that

(3.3)

d dt

Z

M

f²e(A, φ) dvg+ Z

M

f² |∂tA|²+ |∂tφ|²
dvg = −2 Z

M

f hFA, df ∧ ∂tAi + f (DAφ, ∂tφ df ) dvg. In one way, by Young’s inequality, one knows from (3.3) that

d dt

Z

M

f²e(A, φ) dvg≤ C R⁻² Z

M

e(A, φ) dvg. Integrate the above inequality from S to T and apply Proposition 2.7. One has

Z

M

f²e(A, φ) dv_{g    T}

≤ Z

M

f²e(A, φ) dv_{g    S}

+ C E(S) R⁻²(T − S).

Notice the choice of the cut-off function f . We know that (3.1) holds. In another way, still by (3.3), we have d

dt Z

M

f²e(A, φ) dvg+ C Z

M

f² |∂tA|²+ |∂tφ|²
dvg≥ −C R⁻² Z

M

e(A, φ) dvg.

Same as the derivation of (3.1), we can integrate the above inequality from S to T . Then (3.2) holds.

III.2.

Proposition 3.2. Suppose that (A, φ) is a regular solution of the gradient flow (1.3). Then (∂_t− ∆_M) e(A, φ) + |∇_AF_A|²+

∇²_Aφ 

2≤ C (|R_M| + |F_A|) e(A, φ) + (D_Aν_i(φ), D_Aφ)².

Here ∆_M is the Laplace-Beltrami operator of the manifold M . C > 0 is an universal constant depending only on the geometry of M . R_M is the Riemannian curvature of M .

Proof. In one way, it can be shown that (see [5])

−∆_M |DAφ|² 2



= (∇^∗_A∇ADAφ, DAφ) − |∇A(DAφ)|².

In another way, by making time derivative once and applying the equation (1.3), we have

∂t

 |DAφ|² 2



+ |(glφ, DAφ)|²= − (DA(D_A^∗DAφ) , DAφ) − ((D_A^∗FA) φ, DAφ) + (DAνi(φ), DAφ)². Therefore, sum the above two equalities together,

(∂t− ∆_M) |D_Aφ|² 2



+ |(glφ, DAφ)|²+ |∇A(DAφ)|²=

= (DAνi(φ), DAφ)²− (DA(D^∗_ADAφ) − ∇^∗_A∇ADAφ, DAφ) − ((D^∗_AFA) φ, DAφ) . By Weitzenb¨ock formula, one knows that

D_A(D^∗_AD_Aφ) − ∇^∗_A∇_AD_Aφ = R_M × D_Aφ + F_A× D_Aφ − D_A^∗(F_Aφ).

Moreover, one can also show that

(D^∗_AFA) φ = D^∗_A(FAφ) + ∗(∗FA∧ DAφ).

Therefore,

(∂_t− ∆_M) |D_Aφ|² 2



+ |(g_lφ, D_Aφ)|²+ |∇_A(D_Aφ)|²=

= (D_Aν_i(φ), D_Aφ)²− (R_M × D_Aφ + F_A× D_Aφ, D_Aφ) − (∗ (∗F_A∧ D_Aφ) , D_Aφ) . Obviously, we can bound the right-hand side of the above equality and get

(∂t− ∆_M) |D_Aφ|² 2

 +

∇²_Aφ 

2≤ (DAνi(φ), DAφ)²+ C (|RM| + |FA|) |DAφ|², (3.4)

where C > 0 depends only on the geometry ofM . As for |F_A|², we know that

∆_M |FA|² 2



= − h∇^∗_A∇_AF_A, F_Ai + |∇_AF_A|². Moreover, by the equation (1.3),

∂_t |F_A|² 2



= − hD_A(D^∗_AF_A) , F_Ai − hDA(g_lφ, D_Aφ) g_l, F_Ai . Therefore,

(∂t− ∆_M) |FA|² 2



+ |∇AFA|²= − hDAD^∗_AFA− ∇^∗_A∇AFA, FAi − hDA(glφ, DAφ) gl, FAi . Apply Bianchi’s identity, we have

DAD_A^∗FA− ∇^∗_A∇AFA= R_M × FA+ FA× FA.

Hence, for suitably large constant C, (∂t− ∆_M) |FA|²

2



+ |∇AFA|²≤ C (|R_M| + |FA|) |FA|²− hDA(glφ, DAφ) gl, FAi . Note that

hFA, DA(glφ, DAφ) gli = 2 hFA, (glDAφ, DAφ) gli + |FAφ|². Therefore, one may imply that

(∂_t− ∆_M) |FA|² 2



+ |∇_AF_A|²≤ C (|R_M| + |F_A|) e(A, φ). (3.5) The proof is then completed by summing (3.4) with (3.5).

III.3.

We study an -regularity in this section. In the following, for a given r > 0 and z₀ = (x₀, t₀) ∈ M × R, P_r(z₀) denotes the cylinder

Pr(z0) =(x, t) ∈M × R : x ∈ Br(x0), t0− r²≤ t < t0 . If z0= 0, we simply denote Pr(0) by Pr. Now we state our -regularity as follows.

Proposition 3.3 (-regularity). There exist two positive constants δ0= δ0(m,M ) and 0= 0(m,M , N ) such that if for some

R0∈

0, minn

i(M ), T0^1/2

o

, we have

sup

T₀−R²₀≤t<T₀

Z

B_R0(x0)

e(A(t), φ(t)) dvg< 0, (3.6)

then

sup

P_R0/3(x0,T0)

e(A, φ) ≤ 36 δ₀R⁻²₀ .

Proof. Our proof follows [5] and [18] with some modifications. For convenience, we divide our arguments into four steps shown below.

Step 1. Choose tn ↑ T0 and denote the point (x0, tn) by zn. Obviously, we have P_R0/2(zn) ⊂ PR₀(z0) when n is suitably large. Here z0= (x0, T0). Let rn ∈ [R0/4, R0/2] such that

(R0/2 − rn)² sup

P_rn(zn)

e(A, φ) = max

R₀/4≤r≤R₀/2 (R0/2 − r)² sup

Pr(zn)

e(A, φ)

! .

Choose z_n^∗∈ P_rn(z_n) such that

en:= e(A, φ)(z_n^∗) = sup

P_rn(z_n)

e(A, φ).

If for some δ0> 0, we have

e_n≤ δ0 (R₀/2 − r_n)⁻², (3.7)

then

(R0/2 − R0/3)² sup

P_R0/3(z_n)

e(A, φ) ≤ (R0/2 − rn)²en≤ δ0.

Moreover,

sup

P_R0/3(zn)

e(A, φ) ≤ 36 δ0R⁻²₀ . (3.8)

If (3.8) holds for any n suitably large, then the proof can be completed by taking n → ∞. In the following, we show that there are δ0> 0 and 0> 0 such that when (3.6) holds, (3.7) is true for any n suitably large.

Furthermore, (3.8) holds for all n suitably large.

Step 2. If on the contrary that (3.7) fails for some n suitably large. Then γn := δ0e⁻¹_n
¹₂

/2 < (R0/2 − rn) /2.

Clearly, one may imply that

Pγ_n(z^∗_n) ⊂ P_(rn+R0/2)/2(zn). (3.9) Rescale (A, φ) in P_γn(z_n^∗) by

An= γnA x^∗_n+ γny, t^∗_n+ γ_n²s
, φn= φ x^∗_n+ γny, t^∗_n+ γ_n²s
, (y, s) ∈ P1, where z_n^∗= (x^∗_n, t^∗_n). The metric in B₁(0) is induced from g in B_γn(x^∗_n) by

gn,ij(y) = gij(x^∗_n+ γny), ∀ y ∈ B1(0).

On P1, we define

H_n = γ_n⁻²|F_An|²+ | D_Anφ_n|². It is known by the above definitions that

Hn(0, 0) ≥ 2 γ_n²(R0/2 − rn)⁻²



R0/2 −r_n+ R₀/2 2

2

sup

P(rn+R0/2)/2(zn)

e(A, φ).

Notice (3.9), the definition of γn and the rescaling (An, φn), one may imply that sup

P₁

H_n≤ 2 δ0. (3.10)

Step 3. Fix s0∈ [−1, 0]. By (3.10) and the regularity of the flow, we have sup

B₁

|FAn|²(·, s₀) ≤ 2 δ₀γ²_n≤ δ0i(M )². (3.11)

Choose a positive constant κ(m) according to Theorem 1.3 in [19] and set δ₀i(M )²= κ(m). It is clear that when κ(m) is suitably small, we can then find a smooth gauge transformation S(s₀) such that d + A_n(·, s₀) is gauge equivalent to a connection d + A^cg_n(·, s₀) which satisfies the Coulomb gauge condition and can be estimated for all p > m as follows:

kA^cg_n(·, s0)k_W1,p(B₁)≤ c(m)

F_{An(·,s0) Lp(B}

1). (3.12)

Let Os₀ be a neighborhood of s0 in [−1, 0]. For any s ∈ Os₀, we act S(s0) on the connection d + An(·, s).

We denote by d + A^cg_n(·, s), s ∈ Os₀, the gauge equivalent connection. Note that even though we put ”cg” as a superscript in the gauge equivalent connection, but one should notice that usually only when s = s0, the connection is in Coulomb gauge. By the regularity of the original gradient flow (1.3), we can assume that the length of Os₀ is small enough such that

sup

s∈O_s0

kA^cg_n(·, s)k_L∞(B₁)≤ kA^cg_n(·, s₀)k_L∞(B₁)+ 1.

Notice (3.11)-(3.12), we then have by Sobolev embedding theorem that sup

s∈O_s0

kA^cg_n(·, s)k_L∞(B₁)≤ c(m),

where c(m) > 0 is a suitably large constant depending on m. It is clear that {Os₀: s0∈ [−1, 0]}

forms a covering of [−1, 0]. Therefore, we can find a set of finite neighborhoods {Os_i} to cover [−1, 0] and maxi sup

s∈O_si

kA^cg_n(·, s)k_L∞(B₁)≤ c(m). (3.13)

Step 4. By the rescaling in Step 2, we know that in P1,

∂sHn− ∆g_nHn= 2 γ_n⁴ (∂t− ∆_M) e(A, φ).

Apply the Bochner-typer inequality in Proposition 3.2 and (3.11), we have

∂_sH_n− ∆_gnH_n ≤ C_m,MH_n+ 2 (D_Anν_i(φ_n), D_Anφ_n)².

Fix an Os_i in Step 3 and notice that the above inequality is gauge invariant. Therefore,

∂sHn− ∆g_nHn≤ Cm,MHn+ 2 D_A^cg_nνi(φ^cg_n ), D_A^cg_nφ^cg_n
² . Notice (3.13). We know that there is a positive constant Cm,M ,N such that

∂sHn− ∆g_nHn≤ Cm,M ,NHn, in Os_i× B1, ∀ i.

Apply parabolic Harnack inequality (see Theorem 6.17 in [8]). We have δ₀/2 = H_n(0, 0) ≤ C_{m,M ,N}

Z

P1

H_n≤ C_{m,M ,N}γ_n⁻² Z

P_γn(z^∗_n)

e(A, φ) dv_gdt. (3.14)

Since P_γn(z^∗_n) ⊂ P_R0(z₀), one may imply from (3.14) that δ₀/2 ≤ C_{m,M ,N} sup

T₀−R²₀≤t<T0

Z

B_R0(x₀)

e(A(t), φ(t)) dv_g≤ 0C_{m,M ,N}.

Therefore, when we choose 0 small enough, then (3.14) fails. In other words, (3.7) holds for any n suitably large, where δ0is determined in Step 3. The proof is then finished.