Stability and energy identity for Yang-Mills-Higgs pairs

TL;DR

This paper investigates the properties and energy identities of Yang-Mills-Higgs pairs, establishing conditions for stability, curvature vanishing, and convergence behavior on high-dimensional and 4-dimensional manifolds.

Contribution

It proves that stable Yang-Mills-Higgs pairs on spheres have Higgs fields of norm 1 and are Yang-Mills, with curvature vanishing in higher dimensions, and establishes an energy identity for sequences on 4-manifolds.

Findings

Higgs field norm equals 1 for stable pairs on S^n, n > 3

Curvature vanishes for n > 4

Energy identity for sequences on 4-manifolds

Abstract

In this paper, we study the properties of the critical points of Yang-Mills-Higgs functional, which are called Yang-Mills-Higgs pairs. We first consider the properties of weakly stable Yang-Mills-Higgs pairs on a vector bundle over S^n (n > 3). When n > 3, we prove that the norm of its Higgs field is 1 and the connection is actually Yang-Mills. More precisely, its curvature vanishes when n > 4. We also use the bubble-neck decomposition to prove the energy identity of a sequence of Yang-Mills-Higgs pairs over a 4-dimensional compact manifold with uniformly bounded energy. We show there is a subsequence converges smoothly to a Yang-Mills-Higgs pair up to gauge modulo finitely many 4-dimensional spheres with Yang-Mills connections.

Equations

$$YM\left(\nabla \right)={\int }_M|R^{\nabla }{|}^{2}dV$$

$${\delta }^{\nabla }{R}^{\nabla }=0.$$

$${\frac{d^2}{d{t}^2}YM\left({\nabla }^t\right)|}_{t=0}\ge 0$$

$$E_{\epsilon }\left(u\right)={\int }_M\left(\frac{{\left|\nabla u\right|}^2}{2}+\frac{{\left(1-{\left|u\right|}^2\right)}^2}{4{\epsilon }^2}\right)dV$$

$$\mathcal{A}(\nabla ,u)=\frac{1}{2}{\int }_M|R^{▽}{|}^2+|d^{▽}u{|}^2+\frac{\lambda }{4}(1-|u{|}^2{)}^{2}dV$$

$$\begin{array}{rl}{\delta }^{▽}{R}^{▽}& =-\frac{1}{2}(d^{▽}u\otimes u^{\ast }-u\otimes (d^{▽}u{)}^{\ast }),\\ {\delta }^{▽}{d}^{▽}u& =\frac{\lambda }{2}(1-|u{|}^2)u,\end{array}$$

$${\frac{d^2}{d{t}^2}\mathcal{A}({\nabla }^t,u^t)|}_{t=0}\ge 0.$$

$$\mathcal{A}(\nabla ,\Phi )=\frac{1}{2}{\int }_M|R^{▽}{|}^2+|d^{▽}\Phi {|}^2+\frac{\lambda }{4}(1-|\Phi {|}^2{)}^{2}dV,$$

$$\begin{array}{rl}{\delta }^{▽}{R}^{▽}& =[d^{▽}\Phi ,\Phi ],\\ {\delta }^{▽}{d}^{▽}\Phi & =\frac{\lambda }{2}(1-|\Phi {|}^2)\Phi .\end{array}$$

$$\underset{i\to \infty }{\lim }\mathcal{A}({\nabla }_i,u_i)=\mathcal{A}({\nabla }_{\infty },u_{\infty })+\sum _{j=1}^l\sum _{k=1}^{K_j}YM({\tilde{\nabla }}_{jk}).$$

$$\nabla =d+A$$

$$R^{\nabla }=dA+\frac{1}{2}[A\wedge A]$$

$$[\phi \wedge \psi {]}_{X,Y}=\left[{\phi }_X,{\psi }_Y\right]-\left[{\phi }_Y,{\psi }_X\right]$$

$$\nabla (\varphi )=[\nabla ,\varphi ],$$

$${\nabla }_X(\frac{1}{2}(u\otimes {\varphi }^{\ast }-\varphi \otimes u^{\ast }))=\frac{1}{2}({\nabla }_{X}u\otimes {\varphi }^{\ast }-\varphi \otimes ({\nabla }_{X}u{)}^{\ast })+\frac{1}{2}(u\otimes ({\nabla }_X\varphi {)}^{\ast }-{\nabla }_X\varphi \otimes u^{\ast }).$$

$$R_{X,Y}^{\nabla }(\varphi )=[R_{X,Y}^{\nabla },\varphi ],$$

$$d^{\nabla }(\alpha \otimes \sigma )=(d\alpha )\wedge \sigma +(-1{)}^p\alpha \otimes \nabla \sigma$$

$$⟨\varphi ,\phi ⟩=\frac{1}{2}\mathrm{Tr}\left({\varphi }^T\phi \right),\text{ where }\phi ,\varphi \in {\Omega }^0\left({\mathfrak{g}}_E\right).$$

$$⟨[\varphi ,\phi ],\rho ⟩=⟨\varphi ,[\phi ,\rho ]⟩.$$

$$⟨\varphi ,\phi ⟩=\frac{1}{p!}\sum _{1\le i_1,...,i_p\le n}⟨\varphi (e_{i_1},...e_{i_p}),\phi (e_{i_1},...,e_{i_p})⟩$$

$$\left(\phi ,\psi \right)={\int }_M⟨\phi ,\psi ⟩dV,\text{ for any }\phi ,\psi \in {\Omega }^p\left({\mathfrak{g}}_E\right).$$

$$(d^{\nabla }\varphi {)}_{X_0,\cdots ,X_p}=\sum _{k=0}^p(-1{)}^k({\nabla }_{X_k}\varphi {)}_{X_0,\cdots ,\widehat{X_k},\cdots ,X_p},$$

$$({\delta }^{\nabla }\varphi {)}_{X_1,\cdots ,X_{p-1}}=-\sum _{j=1}^n({\nabla }_{e_j}\varphi {)}_{e_j,X_1,\cdots ,X_{p-1}},$$

$${\Delta }^{▽}=d^{▽}{\delta }^{▽}+{\delta }^{▽}{d}^{▽}$$

$${\nabla }^{\ast }\nabla =-\sum _j({\nabla }_{e_j}{\nabla }_{e_j}-{\nabla }_{D_{e_j}{e}_j}).$$

$${\mathfrak{R}}^{▽}(\psi )(X)=\sum _j[R^{▽}(e_j,X),\psi (e_j)],$$

$${\mathfrak{R}}^{▽}(\phi )(X,Y)=\sum _j[R^{▽}(e_j,X),\phi (e_j,Y)]-[R^{▽}(e_j,Y),\phi (e_j,X)].$$

$${\Delta }^{▽}\psi ={\nabla }^{\ast }\nabla \psi +\psi \circ \mathrm{Ric}+{\mathfrak{R}}^{▽}(\psi ),$$

$${\Delta }^{▽}\phi ={\nabla }^{\ast }\nabla \phi +\phi \circ (\mathrm{Ric}\wedge \mathrm{Id}+2R_M)+{\mathfrak{R}}^{▽}(\phi ),$$

$$\mathrm{Ric}\left(X\right)=\sum _{j}R(X,e_j)e_j,$$

$${\left(\mathrm{Ric}\wedge \mathrm{Id}\right)}_{X,Y}=\left(\mathrm{Ric}\wedge \mathrm{Id}\right)\left(X\wedge Y\right)=\mathrm{Ric}(X)\wedge Y+X\wedge \mathrm{Ric}(Y),$$

Full text

Stability and energy identity for Yang-Mills-Higgs pairs

Xiaoli Han1

Xishen Jin2

Yang Wen${}^{3^{*}}$

1Math department of Tsinghua university, Beijing, 100084, China

hanxiaoli@mail.tsinghua.edu.cn

2School of Mathematics, Remin University of China, Beijing, 100872, China

jinxishen@ruc.edu.cn

${}^{3^{*}}$Academy of Mathematics and Systems Science, The Chinese Academy of Science, Beijing, 100190, China

wenyang19@mails.ucas.ac.cn

∗Corresponding author.

**Abstract:

††footnotetext: This is the revised version submitted on 2023.1.7In this paper, we study the properties of the critical points of Yang-Mills-Higgs functional, which are called Yang-Mills-Higgs pairs. We first consider the properties of weakly stable Yang-Mills-Higgs pairs on a vector bundle over $S^{n}$ $(n\geq 4)$. When $n\geq 4$, we prove that the norm of its Higgs field is $1$ and the connection is actually Yang-Mills. More precisely, its curvature vanishes when $n\geq 5$. We also use the bubble-neck decomposition to prove the energy identity of a sequence of Yang-Mills-Higgs pairs over a $4-$dimensional compact manifold with uniformly bounded energy. We show there is a subsequence converges smoothly to a Yang-Mills-Higgs pair up to gauge modulo finitely many $4-$dimensional spheres with Yang-Mills connections.**

Keywords: Yang-Mills-Higgs pairs

1 Introduction

Let $\left(M,g\right)$ be an $n$-dimensional closed Riemannian manifold and $E$ be a vector bundle of rank $r$ over $M$ with structure group $G$, where $G$ is a compact Lie group. Let $\mathfrak{g}_{E}$ be the adjoint bundle of $E$. The classical Yang-Mills functional defined on the space of connections of $E$ is given by

[TABLE]

where $\nabla$ is a connection on $E$, $R^{\nabla}$ denotes its curvature and $dV$ is the volume form of $g$. We denote $d^{\nabla}$ to be the exterior differential induced by $\nabla$ and $\delta^{\triangledown}$ be the formal adjoint of $d^{\triangledown}$. The critical points of Yang-Mills functional are called Yang-Mills connections and they satisfy

[TABLE]

Our interests are the Yang-Mills connections which minimize the Yang-Mills functional locally. At such a connection $\nabla$, the second variation of the Yang-Mills functional should be non-negative, i.e.

[TABLE]

where $\nabla^{t}$ is a curve of connections with $\nabla^{0}=\nabla$. Such connections are called stable. Considering the second variation of the Yang-Mills functional with respect to deformations generated by special vector fields, J. Simons announced that every stable Yang-Mills connection on $S^{n}$ is flat, if $n>4$ in Tokyo in September of 1977. Bourguignon-Lawson [1] gave a detailed proof of this result.

The Ginzburg-Landau equations are the Euler-Lagrange equation of the Ginzburg-Landau functional

[TABLE]

where $u$ is a complex-valued function on $M$. In [2], Cheng proved that every stable solutions of the Ginzburg-Landau equation on $S^{n}$ for $n\geq 2$ are the constant with absolute value $1$.

In this paper, we consider the following Yang-Mills-Higgs functional with self-interaction parameter $\lambda\geq 0$ as a combination of the Yang-Mills functional and Ginzburg-Landau functional

[TABLE]

where $u\in\Omega^{0}(E)$ is a Higgs field. The Yang-Mills-Higgs functional on $\mathbb{R}^{3}$ with structure group $SU\left(2\right)$ was first introduced by P. Higgs in [6]. A. Jaffe-C. Taubes [9] extended the Yang-Mills-Higgs functional to $\mathbb{R}^{n}$ and also general manifolds. Its critical point is the so-called magnetic monopole (we also call it a Yang-Mills-Higgs pair), i.e. a pair $\left(\nabla,u\right)$ satisfying

[TABLE]

where for any $u\in\Omega^{0}(E)$ and $\phi\in\Omega^{p}(E)$, $\frac{1}{2}(u\otimes\phi^{\ast}-\phi\otimes u^{\ast})\in\Omega^{p}(\mathfrak{g}_{E})$ such that for any $\varphi\in\Omega^{p}(\mathfrak{g}_{E})$, we have $\langle\frac{1}{2}(u\otimes\phi^{\ast}-\phi\otimes u^{\ast}),\varphi\rangle=-\langle\phi,\varphi u\rangle$.

Similar to the stable Yang-Mills connection, a Yang-Mills-Higgs pair $\left(\nabla,u\right)$ is called stable if for any curve $\left(\nabla^{t},u^{t}\right)$ such that $\nabla^{0}=\nabla$ and $u^{0}=u$, there holds

[TABLE]

Furthermore, we can define the notation of the weakly stable of Yang-Mills-Higgs pairs (c.f. Definition 2.1). The purpose of the present work is to extend some of the results of J. Simons and Bourguignon-Lawson [1] about weakly stable Yang-Mills connections on $S^{n}$ to the weakly stable Yang-Mills-Higgs pairs. We prove the following theorem.

Theorem 1.1**.**

Assume $(\nabla,u)$ is a weakly stable Yang-Mills-Higgs pairs on $S^{n}$ , then

If $n\geq 5$, then $R^{\triangledown}=0$, $d^{\triangledown}u=0$ and $|u|=1$. 2. 2.

If $n=4$, then $d^{\triangledown}u=0$, $|u|=1$ and $\nabla$ is a Yang-Mills connection (i.e. $\delta^{\triangledown}R^{\triangledown}=0$).

The Higgs fields taking values in $\Omega^{0}(\mathfrak{g}_{E})$ are sometimes concerned in some physical research. The corresponding Yang-Mills-Higgs functional is

[TABLE]

where $\Phi\in\Omega^{0}(\mathfrak{g}_{E})$. The Euler-Lagrange equation of $\mathscr{A}$ is

[TABLE]

The stable Yang-Mills-Higgs pairs $(\nabla,\Phi)$ on $S^{n}$ have similar properties, which are discussed in Section 4.

The energy identity was first established in [3] in dimension $4$ manifolds for sequences of anti-self-dual Yang-Mills fields. In [16], Tian proved that the defect measure of sequences of Yang-Mills fields on a Riemannian manifold $(M,g)$ of dimension $n$ ($n\geq 4$) is carried by a $n-4$-rectifiable subset $S$ of $M$. Riviére [13] proved that, in $4$ dimension, at any point of $S$ it is the sum of $L^{2}$ energies of Yang-Mills fields on $S^{4}$ and this result holds on any dimension under the additional assumption on the $W^{2,1}$ norm of the curvature. Moreover, in [11], Nabor-Valtorta proved that the $W^{2,1}$-norm is automatically bounded for a sequence of stationary Yang-Mills fields with bounded energy.

In [14], Song proved the energy identity for a sequence of Yang-Mills-Higgs pairs on a fiber bundle with curved fiber spaces over a compact Riemannian surface and the blow-up only occurs in the Higgs part in the 2-dimensional case. In Section 5, we assume that $M$ is a 4-dimensional compact Riemannian manifold and $\{(\nabla_{i},u_{i})\}$ is a sequence of Yang-Mills-Higgs pairs over $M$ with uniformly bounded energy $\mathscr{A}(\nabla_{i},u_{i})\leq K$. Unlike the case of 2-dimensional manifolds, it will be shown that there is no energy concentration point for Higgs field over $4-$dim manifolds and the blow-up only occurs in the curvature part. We prove the following theorem.

Theorem 1.2**.**

Assume $\{(\nabla_{i},u_{i})\}$ is a family of Yang-Mills-Higgs pairs which satisfy the equation (4) and $\mathscr{A}(\nabla_{i},u_{i})\leq K$. Then there is a finite subset $\Sigma=\{x_{1},...,x_{l}\}\subset M$, a Yang-Mills-Higgs pair $(\nabla_{\infty},u_{\infty})$ on $M\setminus\Sigma$ and Yang-Mills connections $\{\widetilde{\triangledown}_{jk}\mid 1\leq j\leq l,1\leq k\leq K_{j}\}$ over $S^{4}$, such that there is a subsequence of $\{(\nabla_{i},u_{i})\}$ converges to $(\nabla_{\infty},u_{\infty})$ in $C^{\infty}_{loc}(M\setminus\Sigma)$ under gauge transformations and

[TABLE]

2 Preliminary

Let $\left(M,g\right)$ be a compact Riemannian manifold and $D$ be its Levi-Civita connection. Let $E\to M$ be a rank $r$ vector bundle over $M$ with a compact Lie group $G\subset SO\left(r\right)$ as its structure group. We also assume $\langle\ ,\ \rangle$ is a Riemannian metric of $E$ compatible with the action of $G$ . Let $\mathfrak{g}_{E}$ be the adjoint bundle of $E$. Assume $\nabla:\Omega^{0}(E)\to\Omega^{1}(E)$ is a connection on $E$ compatible with the metric $\langle\ ,\ \rangle$. Locally, $\nabla$ takes the form

[TABLE]

where $A\in\Omega^{1}\left(\mathfrak{g}_{E}\right)$ is the connection $1$-form.

For any connection $\nabla$ of $E$, the curvature $R^{\nabla}=\nabla^{2}\in\Omega^{2}\left(\mathfrak{g}_{E}\right)$ measures the extent to which $\nabla$ fails to commute. Locally, $R^{\nabla}$ is given by

[TABLE]

where the bracket of $\mathfrak{g}_{E}$-valued $1$-forms $\varphi$ and $\psi$ is defined to be

[TABLE]

as in [1].

The connection $\nabla$ on $E$ induces a natural connection on $\mathfrak{g}_{E}$. Indeed for $\phi\in\Omega^{0}(\mathfrak{g}_{E})$ we define

[TABLE]

i.e. $\nabla(\phi)(\sigma)=\nabla(\phi(\sigma))-\phi(\nabla\sigma)$ for any section $\sigma$ of $E$. By direct calculation, for any $X\in T(M)$ we have

[TABLE]

Similarly, the curvature of this connection on $\mathfrak{g}_{E}$ is given by the formula

[TABLE]

where $R^{\nabla}$ on the right denotes the curvature of $E$. We define an exterior differential $d^{\nabla}:\Omega^{p}(\mathfrak{g}_{E})\rightarrow\Omega^{p+1}(\mathfrak{g}_{E}),p\geq 0$ as follows. For each real-valued differential $p$-form $\alpha$ and each section $\sigma$ of $E$, we set

[TABLE]

and extend the definition to general $\psi\in\Omega^{p}(E)$ by linearity. Note that $d^{\nabla}=\nabla$ on $\Omega^{0}(E)$ and $(d^{\nabla}(d^{\nabla}\sigma))_{X,Y}=R^{\nabla}_{X,Y}(\sigma)$.

The inner product of $\Omega^{0}\left(\mathfrak{g}_{E}\right)$ induced by the trace inner product metric on $\mathfrak{so}\left(r\right)$ is given by

[TABLE]

Then for any $\phi,\varphi,\rho\in\Omega^{0}(\mathfrak{g}_{E})$, we have

[TABLE]

Combining with Riemannian metric $g$, the inner product of $\Omega^{p}(\mathfrak{g}_{E})$ can be defined by

[TABLE]

where $\phi,\varphi\in\Omega^{p}(\mathfrak{g}_{E})$ and $\{e_{i}\mid 1\leq i\leq n\}$ is an orthogonal basis of $TM$. Integrating the inner product $\langle\ ,\ \rangle$ over $M$, we get a global inner product $\left(\ ,\ \right)$ in $\Omega^{p}\left(\mathfrak{g}_{E}\right)$, i.e.

[TABLE]

Define the operator $\delta^{\nabla}:\Omega^{p+1}(\mathfrak{g}_{E})\rightarrow\Omega^{p}(\mathfrak{g}_{E}),p\geq 0,$ to be the formal adjoint of the operator $d^{\nabla}$. In local coordinates, for any $\phi\in\Omega^{p}(\mathfrak{g}_{E})$,

[TABLE]

[TABLE]

where $\{e_{i}\}$ is an orthonormal basis of $TM$.

We can define the Laplace-Beltrami operator $\Delta^{\nabla}$ by

[TABLE]

and the rough Laplacian operator $\nabla^{*}\nabla$ by

[TABLE]

For $\psi\in\Omega^{1}(\mathfrak{g}_{E})$ and $\varphi\in\Omega^{2}(\mathfrak{g}_{E})$, we recall the following operator $\mathfrak{R}^{\triangledown}$ defined in [1]

[TABLE]

Then we have the following Bochner-Weizenböck formula first introduced in [1].

Theorem 2.1**.**

For any $\psi\in\Omega^{1}(\mathfrak{g}_{E})$ and $\varphi\in\Omega^{2}(\mathfrak{g}_{E})$, we have

[TABLE]

where

• •

$\operatorname{Ric}:TM\to TM$* is the Ricci transformation defined by*

[TABLE]

• •

$\psi\circ\operatorname{Ric}\in\Omega^{1}\left(\mathfrak{g}_{E}\right)$* and $\left(\psi\circ\operatorname{Ric}\right)_{X}=\psi_{\operatorname{Ric}\left(X\right)}\in\Omega^{0}\left(\mathfrak{g}_{E}\right),$*

• •

$\operatorname{Ric}\wedge\operatorname{Id}$* is the extension of the Ricci transformation $\operatorname{Ric}$ to $\wedge^{2}TM$ given by*

[TABLE]

• •

$R_{M}$* is the curvature of $TM$,*

• •

For any map $\omega:\wedge^{2}TM\to\wedge^{2}TM$, the composite map $\varphi\circ\omega:\wedge^{2}TM\to\Omega^{0}\left(\mathfrak{g}_{E}\right)$ is defined by

[TABLE]

Remark 2.1**.**

In particular, on the standard sphere $S^{n}$,

[TABLE]

and

[TABLE]

Thus, we have

[TABLE]

and

[TABLE]

Note that for any $B\in\Omega^{1}\left(\mathfrak{g}_{E}\right)$ and $w\in\Omega^{0}\left(E\right)$,

[TABLE]

and

[TABLE]

Assume $(\nabla,u)$ is a Yang-Mills-Higgs pair satisfying the equation (2) and $(\nabla^{t},u^{t})$ is a curve on

[TABLE]

such that $\nabla^{0}=\nabla$ and $u^{0}=u$. If we assume $\frac{d}{dt}(\nabla^{t},u^{t})\mid_{t=0}=(B,w)$, the second variation of $\mathscr{A}$ is

[TABLE]

For any gauge transformation $g\in\mathcal{G}$, $g$ acts on $(\nabla,u)$ such that $(\nabla^{g},u^{g})=(g\circ\nabla\circ g^{-1},gu)$. Then for any $\sigma\in\Omega^{0}(\mathfrak{g}_{E})$, let $g_{t}=\exp(t\sigma)$ be a family of gauge transformations, the variation of $(\nabla,u)$ along $\sigma$ is

[TABLE]

In addition, the Yang-Mills-Higgs functional $\mathscr{A}$ is invariant under the action of gauge group. So it is interesting to consider the variation $\left(B,w\right)$ perpendicular to the direction of gauge transformation with respect to the global inner product on $\Omega^{1}\left(\mathfrak{g}_{E}\right)\times\Omega^{0}\left(\mathfrak{g}_{E}\right)$. Define

[TABLE]

then $(B,w)\in\operatorname{Im}(\zeta)^{\perp}$ if and only if for any $\sigma$,

[TABLE]

Thus the space of admissible variation at $\left(\nabla,u\right)$ is

[TABLE]

Using $\delta^{\triangledown}(Bu)=(\delta^{\triangledown}B)u-B\llcorner d^{\triangledown}u$, where

[TABLE]

and for $(B,w)\in\mathcal{C}$, we have

[TABLE]

For any $x\in M$, let $\{e_{i}\mid 1\leq i\leq n\}$ be an orthogonal basis of $T_{x}M$. Since $B(e_{i})\in\mathfrak{so}(E_{x})$, we have

[TABLE]

Hence for $(B,w)\in\mathcal{C}$, the second variation of $\mathscr{A}$ is

[TABLE]

Define an operator

[TABLE]

where

[TABLE]

It is easy to see that $\mathscr{S}^{(\nabla,u)}$ is a self-adjoint operator on $\Omega^{1}(\mathfrak{g}_{E})\times\Omega^{0}(E)$. Furthermore, we can prove that $\mathscr{S}^{(\nabla,u)}$ is a self-adjoint operator on $\mathcal{C}$.

Lemma 2.2**.**

$\mathscr{S}^{(\nabla,u)}(\mathcal{C})\subset\mathcal{C}$*. *

Proof**.**

Denote $\mathscr{S}^{(\nabla,u)}=(\mathscr{S}_{1},\mathscr{S}_{2})$. We only need to prove that for any $(B,w)\in\mathcal{C}$ and $\sigma\in\Omega^{0}(\mathfrak{g}_{E})$, we have

[TABLE]

First we have

[TABLE]

The equation (2) implies that for any $\phi\in\Omega^{1}(\mathfrak{g}_{E})$, we have

[TABLE]

By direct calculation, we have

[TABLE]

Thus

[TABLE]

On the other hand, $\sigma\in\Omega^{0}(\mathfrak{g}_{E})$ implies $\langle u,\sigma u\rangle=0$. By the equation (2) we have

[TABLE]

Hence by $(B,w)\in\mathcal{C}$, we have

[TABLE]

Then we have $\int_{M}\langle\mathscr{S}^{(\nabla,u)}(d^{\triangledown}\sigma,-\sigma u),(B,w)\rangle dV=0$ and we finish the proof.

Then $\mathscr{S}^{(\nabla,u)}\mid_{\mathcal{C}}:\mathcal{C}\to\mathcal{C}$ is a self adjoint and elliptic operator. The eigenvalues of $\mathscr{S}^{(\nabla,u)}$ are given by

[TABLE]

Similar as in [1], we define the weakly stability of Yang-Mills-Higgs functional at $\left(\nabla,u\right)$ as following.

Definition 2.1**.**

*Assume $(\nabla,u)$ satisfies (2), then it is called weakly stable if $\lambda_{1}\geq 0$ and stable if $\lambda_{1}>0$. *

3 Stability of Yang-Mills-Higgs pairs on $S^{n}$

In this section, we prove Theorem 1.1. Now, we assume $\left(M,g\right)$ is the standard Euclidean sphere $S^{n}$. In [10], the conformal Killing vector fields of $S^{n}$ play an important role in studying the non-existence of stable varifolds or currents. Similar methods have been applied to study weakly stable Yang-Mills connections on $S^{n}$ in [1]. The conformal Killing vector fields of $S^{n}$ are the gradients of eigenfunctions corresponding to the first non-zero eigenvalue of the Laplace operator. Let us summarize the properties of these vector fields as follow.

Proposition 3.1**.**

For any $v=(v^{1},...,v^{n+1})\in\mathbb{R}^{n+1}$, let $F_{v}(x):\mathbb{R}^{n+1}\to\mathbb{R}$, $x\mapsto v\cdot x$ be the inner product of $v$ and $x$. Define $f_{v}=F_{v}\mid_{S^{n}}$ to be the restriction of $F_{v}$ to $S^{n}$. Then

[TABLE]

is a conformal Killing vector fields on $S^{n}$ and satisfies

1. (1).

$D_{X}V=-f_{v}X,$** 2. (2).

$D^{\ast}DV=V.$**

Remark 3.1**.**

In fact, the space $\mathscr{V}$ of all $V$ defined above is the orthogonal complement to the Killing vector fields in the space of all conformal vector fields on $S^{n}$, i.e.

[TABLE]

Similar as in [1], we choose the variation of connection to be $B_{v}=i_{V}R^{\triangledown}$, where $i_{V}$ is the contraction about $V$. The corresponding variation of the Higgs field $w_{v}$ satisfies

[TABLE]

Hence, $w_{v}=\nabla_{V}u$ satisfies $(B_{v},w_{v})\in\mathcal{C}$. If we define a quadratic form $\mathscr{L}^{(\nabla,u)}$ on $\mathcal{C}$ by setting

[TABLE]

then $\left.\frac{d^{2}}{dt^{2}}\mathscr{A}(\nabla^{t},u^{t})\right|_{t=0}=\mathscr{L}^{(\nabla,u)}(B,w)$ for any $(B,w)\in\mathcal{C}$. $Q(v_{1},v_{2})=\mathscr{L}^{(\nabla,u)}(B_{v_{1}},w_{v_{2}})$ can be viewed as a quadratic form.

Lemma 3.2**.**

Assume $(\nabla,u)$ is a Yang-Mills-Higgs pair on $S^{n}$. Then for any $v\in\mathbb{R}^{n+1}$, we have

[TABLE]

Proof**.**

Denote $\mathscr{S}^{(\nabla,u)}=(\mathscr{S}_{1},\mathscr{S}_{2})$. For any $x\in S^{n}$, let $\{e_{i}\mid 1\leq i\leq n\}$ be a local orthonormal frame of $S^{n}$ near $x$. According to (9), we have

[TABLE]

Since $De_{k}(x)=0$, at $x$

[TABLE]

where we have used (1) of Proposition 3.1. By $2\mathfrak{R}^{\triangledown}(i_{V}R^{\triangledown})(e_{k})-\mathfrak{R}^{\triangledown}(R^{\triangledown})(V,e_{k})=0$, we have

[TABLE]

Similarly, by the definition of curvature and $[X,Y]=D_{X}Y-D_{Y}X$, we have

[TABLE]

At $x$, we have

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where we use $\sum_{j}R(e_{j},X)e_{j}=(1-n)X$ on $S^{n}$. Hence

[TABLE]

By equation (2), at $x$ we have

[TABLE]

And

[TABLE]

Note that at $x$, by the equation (2) we have

[TABLE]

Hence by $R^{\triangledown}(V,e_{k})u=\nabla_{V}\nabla_{e_{k}}u-\nabla_{e_{k}}\nabla_{V}u-\nabla_{[V,e_{k}]}u$, we have

[TABLE]

And thus

[TABLE]

On the other hand, by equation (2) and note that $-2\mathfrak{R}^{\triangledown}(d^{\triangledown}u)(V)-2i_{V}R^{\triangledown}\llcorner d^{\triangledown}u=0$, we have

[TABLE]

Then we finish the proof.

Lemma 3.3**.**

Let $\{v_{i}\mid 1\leq i\leq n+1\}\in\mathbb{R}^{n+1}$ be an orthogonal vector in $\mathbb{R}^{n+1}$ and the corresponding Killing field is $V_{i}$. Then

[TABLE]

Proof**.**

Assume $v_{i}$ is the vector in $\mathbb{R}^{n+1}$ such that the $i$-th component is $1$ and the others are [math], then at $x=(x^{1},...,x^{n+1})$, we have $f_{v_{i}}(x)=x^{i}$ and

[TABLE]

Then according to Lemma 3.2,

[TABLE]

where at any $x\in S^{n}$, $q$ is a quadratic form on $\mathbb{R}^{n+1}$ defined by

[TABLE]

and $V,W\in TS^{n}$ with respect to $v,w\in\mathbb{R}^{n+1}$ defined in Proposition 3.1.

Now, we compute the value $q\left(v,w\right)$ at $x\in S^{n}$. Since $q$ is a quadratic form on $\mathbb{R}^{n+1}$, the trace $\sum\limits_{i=1}q\left(v_{i},v_{i}\right)$ is independent of the choice of basis of $\mathbb{R}^{n+1}$. We assume that $\left\{e_{j}\right\}_{j=1}^{n}$ is any orthonormal basis for $T_{x}S^{n}$. Then $\left\{e_{0}=x,e_{1},e_{2},\cdots,e_{n}\right\}$ forms an orthonormal basis for $\mathbb{R}^{n+1}$. In particular, at $x\in S^{n}$, we have

[TABLE]

From Proposition 3.1, at $x$, the Killing vector fields $\varepsilon_{0},\varepsilon_{1},\cdots,\varepsilon_{n}$ with respect to $e_{0},e_{1},e_{2},\cdots,e_{n}$ are

[TABLE]

Since $\left\{e_{j}\right\}_{j=1}^{n}$ forms an orthonormal basis for $T_{x}S^{n}$, we have

[TABLE]

Hence, at $x$,

[TABLE]

and we complete the proof of this lemma.

From the lemma above, we can immediately prove Theorem 1.1 on $S^{n}(n\geq 5)$.

Theorem 3.4**.**

*Assume $(\nabla,u)$ is a Yang-Mills-Higgs pair on $S^{n}$ for $n\geq 5$, then $(\nabla,u)$ is weakly stable if and only if $R^{\triangledown}=0$, $d^{\triangledown}u=0$ and $|u|=1$. *

Proof**.**

If $(\nabla,u)$ satisfies $R^{\triangledown}=0$, $d^{\triangledown}u=0$ and $|u|=1$, then $\mathscr{A}(\nabla,u)=0$ is a minimum of $\mathscr{A}$ and $(\nabla,u)$ is obviously weakly stable.

On the other hand, assume $(\nabla,u)$ is weakly stable, then for any $v\in\mathbb{R}^{n+1}$, $\mathscr{L}^{(\nabla,u)}(i_{V}R^{\triangledown},\nabla_{V}u)\geq 0$. By Lemma 3.3, we have $R^{\triangledown}=0$ and $d^{\triangledown}u=0$. From Equation (2), we have $u=0$ or $|u|=1$. However, $\left(\nabla,0\right)$ can not be weakly stable due to the expression of $\mathscr{A}$. In fact, we choose a nonzero section $w\in\Omega^{0}(E)$ such that $d^{\triangledown}w=0$ and perturb $\left(\nabla,u\right)$ along $\left(0,w\right)\in\mathcal{C}$. Then

[TABLE]

which contradicts with the weakly stable condition. Thus $|u|=1$.

The case when $n=4$ is similar as the proof above. First, we can get $d^{\triangledown}u=0$. Then the Yang-Mills condition can be obtained according to Equation (2).

4 The Higgs fields of $\Omega^{0}(\mathfrak{g}_{E})$

In this section, we consider the Yang-Mills-Higgs functional

[TABLE]

where $\Phi\in\Omega^{0}(\mathfrak{g}_{E})$. Assume that $(\nabla,\Phi)$ is a Yang-Mills-Higgs pair satisfying the equation (4) and $(\nabla^{t},\Phi^{t})$ is a curve on

[TABLE]

such that $\nabla^{0}=\nabla$ and $\Phi^{0}=\Phi$. If we assume $\frac{d}{dt}(\nabla^{t},\Phi^{t})\mid_{t=0}=(B,\phi)$, then the second variation of $\mathscr{A}$ is

[TABLE]

For any gauge transformation $g\in\mathcal{G}$, $g$ acts on $(\nabla,\Phi)$ such that

[TABLE]

Then for any $\sigma\in\Omega^{0}(\mathfrak{g}_{E})$, assume $g_{t}=exp(t\sigma)$ is a family of gauge transformations. The variation direction of $(\nabla,u)$ along $\sigma$ is

[TABLE]

Similar to the case when the Higgs fields take values in $\Omega^{0}(E)$, define

[TABLE]

and $\mathcal{C}=Im(\zeta)^{\perp}$. By direct calculation, we have

[TABLE]

On $\mathcal{C}$, we have

[TABLE]

Then if $\left(B,\phi\right)\in\mathcal{C}$, the second variation formula of $\mathscr{A}$ can be written as

[TABLE]

where $d^{\triangledown}\Phi\llcorner B=\sum\limits_{i=1}^{n}[\nabla_{e_{i}}\Phi,B(e_{i})]$. Hence, we can define an operator

[TABLE]

where

[TABLE]

Lemma 4.1**.**

$\mathscr{S}^{(\nabla,\Phi)}(\mathcal{C})\subset\mathcal{C}$*. *

Proof**.**

Denote $\mathscr{S}^{(\nabla,\Phi)}=(\mathscr{S}_{1},\mathscr{S}_{2})$. Note that $\mathscr{S}^{(\nabla,\Phi)}$ is self-adjoint on $\Omega^{1}(\mathfrak{g}_{E})\times\Omega^{0}(\mathfrak{g}_{E})$, we only need to prove that for any $(B,\phi)\in\mathcal{C}=\operatorname{Im}^{\perp}\left(\zeta\right)$ and $\sigma\in\Omega^{0}(\mathfrak{g}_{E})$, we have

[TABLE]

Since $\zeta\left(\sigma\right)=\left(-d^{\nabla}\sigma,\left[\sigma,\Phi\right]\right)$, we have

[TABLE]

where we use the fact that $\langle\Phi,\left[\sigma,\Phi\right]\rangle=0$ in the last equality. Note that

[TABLE]

we have

[TABLE]

For the third term in $\mathscr{S}_{1}\left(\zeta\left(\sigma\right)\right)$, we have

[TABLE]

where we use the Jacobi identity in the third equality to $\left[\left[\sigma,d^{\nabla}\Phi\right],\Phi\right]$ and apply Equation (4) in the last equality. Combining the equations above, we have

[TABLE]

In the following we deal with $\mathscr{S}_{2}(\zeta(\sigma))$. By direct computation, we have

[TABLE]

Inserting it to $\mathscr{S}_{2}(\zeta(\sigma))$ and applying the second equation in Equation (4), we obtain

[TABLE]

Hence,

[TABLE]

where we use the condition $\left(B,\Phi\right)\in\mathcal{C}$, i.e. $\delta^{\nabla}B=\left[\Phi,\phi\right]$ in the second equality.

Hence $\mathscr{S}^{(\nabla,\Phi)}\mid_{\mathcal{C}}:\mathcal{C}\to\mathcal{C}$ is an elliptic self-adjoint operator. If we assume the minimum eigenvalue of $\mathscr{S}^{(\nabla,u)}\mid_{\mathcal{C}}$ is $\lambda_{1}$, then $(\nabla,u)$ is called weakly stable if $\lambda_{1}\geq 0$ and stable if $\lambda_{1}>0$. For any $(B,\phi)\in\mathcal{C}$, define

[TABLE]

Then $\frac{d^{2}}{dt^{2}}\mathscr{A}(\nabla^{t},u^{t})\mid_{t=0}=\mathscr{L}^{(\nabla,u)}(B,\phi)$.

Now assume $M=S^{n}$ and $(\nabla,\Phi)$ is a Yang-Mills-Higgs pairs satisfying (4). For any $v\in\mathbb{R}^{n+1}$, let $f_{v}(x)=v\cdot x\in C^{\infty}(S^{n})$ and $V=grad(f_{v})$ be the Killing field (17) on $S^{n}$. Similar to the case when the Higgs fields in $\Omega^{0}(E)$, assume $B_{v}=i_{V}R^{\triangledown}$ and $\phi_{v}=\nabla_{V}\Phi$. Then $(B_{v},\phi_{v})\in\mathcal{C}$. Similarly, we have

Lemma 4.2**.**

Assume $(\nabla,\Phi)$ is a Yang-Mills-Higgs pair on $S^{n}$. Then for any $v\in\mathbb{R}^{n+1}$, we have

[TABLE]

Proof**.**

Similar to the proof of Lemma 3.2, we have

[TABLE]

and

[TABLE]

where $\mathfrak{R}^{\triangledown}(d^{\triangledown}\Phi)(V)=\sum_{j}[R^{\triangledown}(e_{j},V),\nabla_{e_{j}}\Phi]$ and $\{e_{1},...,e_{n}\}$ is an orthogonal basis of $TS^{n}$. Furthermore, by the Equation (4),

[TABLE]

Also noting that $d^{\triangledown}\Phi\llcorner i_{V}R^{\triangledown}=\mathfrak{R}^{\triangledown}(d^{\triangledown}\Phi)(V)$, we have

[TABLE]

By Equation (4), we have at $x\in S^{n}$

[TABLE]

Thus the second variation of $\mathscr{A}$ at $\left(\nabla,\Phi\right)$ along $\left(i_{V}R^{\nabla},\nabla_{V}\Phi\right)$ is

[TABLE]

We finish the proof.

For an orthogonal basis $\{v_{i}\mid 1\leq i\leq n+1\}$ of $\mathbb{R}^{n+1}$, we assume $\{V_{i}\}$ is the corresponding Killing fields. We have proved that $\sum_{i}f_{v_{i}}V_{i}=0$. Hence by adding $\mathscr{L}^{(\nabla,\Phi)}(i_{V_{i}}R^{\triangledown},\nabla_{V_{i}}\Phi)$ together, we can prove the following lemma.

Lemma 4.3**.**

Assume $\{v_{i}\}$ and $\{V_{i}\}$ defined as above, then we have

[TABLE]

From the lemma above, we can immediately prove the stability theorem.

Theorem 4.4**.**

*Assume $(\nabla,\Phi)$ is a weakly stable Yang-Mills-Higgs pair on $S^{n}$ for $n\geq 5$, then $R^{\triangledown}=0$, $d^{\triangledown}\Phi=0$ and $|\Phi|\equiv 1$.

And if $n=4$, then $d^{\triangledown}\Phi=0$, $|\Phi|\equiv 1$ and $R^{\triangledown}$ is a Yang-Mills connection. *

5 The energy identity for a sequence of Yang-Mills-Higgs pairs

In this section, we assume $M$ is a four-dimensional compact Riemannian manifold. Let $\{\nabla_{i},u_{i}\}$ be a sequence of Yang-Mills-Higgs pairs satisfying the Yang-Mills-Higgs equation (2) with uniformly bounded energy $\mathscr{A}(\nabla_{i},u_{i})\leq K$.

5.1 There is no energy concentration point for the Higgs field.

Assume $(\nabla,u)$ satisfies the equation (2). In this part, we will give the estimate of $\|u\|_{L^{\infty}}$ and $\|d^{\triangledown}u\|_{L^{2}}$. In fact, we will show that if a sequence of Yang-Mills-Higgs $\{(\nabla_{i},u_{i})\}$ weakly converges to $(\nabla,u)$ in $W^{1,2}(\mathfrak{g}_{E})\times W^{1,2}(E)$, then $u_{i}$ converges to $u$ smoothly.

Lemma 5.1**.**

*Assume $(\nabla,u)$ is a Yang-Mills-Higgs pair, then $\|u\|_{L^{\infty}}\leq 1$. *

Proof**.**

Assume $|u|^{2}$ attains the maximum at $x_{0}\in M$. Then at $x_{0}$, we have

[TABLE]

Hence $|u|^{2}(x_{0})\leq 1$ and this implies the lemma.

Lemma 5.2**.**

Assume $(\nabla,u)$ is a Yang-Mills-Higgs pair and $\mathscr{A}(\nabla,u)\leq K$. Then for any $B_{r}=B_{r}(x_{0})\subset M$, we have

[TABLE]

*where $C=C(K,\lambda)$. *

Proof**.**

Choose a cut off function $\eta$ such that $\eta\mid_{B_{r}}\equiv 1$, $supp(\eta)\subset B_{2r}$ and $|d\eta|\leq Cr^{-1}$. Then by the equation (2), we have

[TABLE]

The above lemmas imply the following theorem immediately.

Theorem 5.3**.**

Assume $(\nabla,u)$ is a Yang-Mills-Higgs pair and $\mathscr{A}(\nabla,u)\leq K$, then for any $x\in M$, we have

[TABLE]

In particular, we have

[TABLE]

5.2 The $\epsilon$-regularity

The $\epsilon$-regularity theorem is proved by Uhlenbeck [17] for Yang-Mills connections, Struwe [15] for Yang-Mills flows and Hong-Fang [4] for Yang-Mills-Higgs flows. For completeness, we first give the proof of the $\epsilon$-regularity of Yang-Mills-Higgs pairs here. Let $i(M)$ be the injective radius of $M$. Then for any $r<i(M)$ and $x\in M$, there is a trivialization $\nabla=d+A$ in $B_{r}(x)$.

Lemma 5.4** ($\epsilon$-regularity).**

Assume $(\nabla,u)$ is a Yang-Mills-Higgs pair. There exists $\epsilon_{0}=\epsilon_{0}(M)$ such that if for some $R<i(M)$ and $x_{0}\in M$,

[TABLE]

then for any $r_{1}<R$, we have

[TABLE]

*where $C=C(M)$. *

Proof**.**

There exists $r_{0}<r_{1}$, such that

[TABLE]

where $D_{r}\left(x\right)=\partial B_{r}\left(x\right)$. Then there exists $x_{1}\in D_{r_{0}}(x_{0})$ such that

[TABLE]

Define $e_{0}=(|R^{\triangledown}|^{2}+|d^{\triangledown}u|^{2})(x_{1})$. We claim that $e_{0}\leq 16(r_{1}-r_{0})^{-4}$. In fact, if we suppose $e_{0}>16(r_{1}-r_{0})^{-4}$, then $\rho_{0}=e_{0}^{-\frac{1}{4}}<\frac{r_{1}-r_{0}}{2}$. Define

[TABLE]

such that $\rho(x)=x_{1}+\rho_{0}x$, and

[TABLE]

Let $\widetilde{\nabla}=d+\tilde{A}$ be a connection of $\rho^{\ast}E$ over $B_{1}(0)$ and $e_{\rho_{0}}=|R^{\tilde{\triangledown}}|^{2}+\rho_{0}^{2}|d^{\tilde{\triangledown}}\tilde{u}|^{2}=\rho_{0}^{4}(|R^{\triangledown}|^{2}+|d^{\triangledown}u|^{2})$. Note that $B_{\rho_{0}}(x_{1})\subset B_{\frac{r_{0}+r_{1}}{2}}(x_{0})$, we have

[TABLE]

which implies $|R^{\tilde{\triangledown}}|^{2}+\rho_{0}^{2}|d^{\tilde{\triangledown}}\tilde{u}|^{2}\leq 16$ on $B_{1}(0)$. By equation (2) and the Bochner formula, we have

[TABLE]

and

[TABLE]

Hence

[TABLE]

By Harnack inequality and note that $B_{\rho_{0}}(x_{1})\subset B_{R}(x_{0})$, we have

[TABLE]

It is a contradiction if $\epsilon_{0}$ small enough and we prove the claim.

Then we have

[TABLE]

where, more precisely, $C=2^{12}$. For any $x_{2}\in B_{\frac{r_{1}}{2}}(x_{0})$, define

[TABLE]

Similarly, $\hat{e}=|R^{\hat{\triangledown}}|^{2}+(\frac{r_{1}}{4})^{2}|d^{\hat{\triangledown}}\hat{u}|^{2}=(\frac{r_{1}}{4})^{4}(|R^{\triangledown}|^{2}+|d^{\triangledown}u|^{2})$ satisfies $\hat{e}\leq C$ and hence $\Delta\hat{e}\leq C\hat{e}$ in $B_{1}(0)$. By Harnack inequality we have

[TABLE]

Then we prove the lemma.

In order to obtain the $\epsilon$-regularity of high order derivatives, we need the following lemma.

Lemma 5.5** ([17], Theorem 1.3).**

*Assume $\nabla=d+A$ is a connection over $B_{1}$. There exists $\kappa$ and $c$ such that if $\|R^{\triangledown}\|_{L^{2}(B_{1})}\leq\kappa$, then there exists a gauge $g$ such that for any $p\geq 2$, there is $d^{\ast}A^{g}=0$ and $\|A^{g}\|_{W^{1,p}(B_{1})}\leq c\|R^{\triangledown}\|_{L^{p}(B_{1})}$, where $\kappa$ and $c$ is only determined by $dim(M)$. *

Remark 5.1

By letting $p\to\infty$ we have $\|A^{g}\|_{L^{\infty}(B_{1})}\leq c\|R^{\triangledown}\|_{L^{\infty}(B_{1})}$. This is consist with Theorem 2.7 in [18].

Lemma 5.6** ($\epsilon$-regularity of high order derivative).**

Assume $(\nabla,u)$ is a Yang-Mills-Higgs pair. There exists $\epsilon_{1}=\epsilon_{1}(M)$ such that if for some $R<i(M)$ and $x_{0}\in M$,

[TABLE]

then there exists a gauge transformation $g$ such that for any $r_{2}<R$ and $k\geq 1$, we have

[TABLE]

Proof**.**

Choose $\epsilon_{1}=\min\{\epsilon_{0},\kappa^{2}\}$ and $R_{1}<R_{0}$. By lemma 5.5, there exists a gauge transformation $g$, such that $d^{\ast}A^{g}=0$. For simplicity, we omit the superscript $g$. Then the equations (2) are

[TABLE]

where $\Delta$ is the covariant Laplacian operator on $M$ and

[TABLE]

By lemma 5.5, we have $\|A\|_{L^{\infty}}\leq C(R_{1})\|R^{\triangledown}\|_{L^{\infty}}$ in $B_{\frac{R_{1}}{2}}(x_{0})$. And by Lemma 5.4, we have

[TABLE]

By $L^{p}$-estimate, for any $R_{2}<R_{1}$ and $p>1$, we have $\|A\|_{W^{2,p}(B_{\frac{1}{2}R_{2}})}+\|u\|_{W^{2,p}(B_{\frac{1}{2}R_{2}})}\leq C(R_{1},R_{2},p)$. By Sobolev’s embedding theorem, $\|A\|_{C^{1}(B_{\frac{1}{2}R_{2}})}+\|u\|_{C^{1}(B_{\frac{1}{2}R_{2}})}\leq C(R_{1},R_{2})$. Differentiating the equation (29) and repeat the above process, we can prove that for any $k\geq 1$ and $R_{k}<R_{k-1}$, we have $\|A\|_{C^{k}(B_{\frac{1}{2}R_{2}})}+\|u\|_{C^{k}(B_{\frac{1}{2}R_{2}})}\leq C(R_{1},R_{2},...,R_{k},k)$. For any $r_{2}<R$, by choosing $R>R_{1}>R_{2}>...>r_{2}$, the lemma is proved.

5.3 The proof of theorem 1.2

Assume $\{(\nabla_{i},u_{i})\}$ is a sequence of Yang-Mills-Higgs pairs with uniformly bounded energy $\mathscr{A}(\nabla_{i},u_{i})\leq K$. Define

[TABLE]

$\mathscr{A}(\nabla_{k},u_{k})\leq K$ implies that $\Sigma$ is finite and the number of elements in $\Sigma$ is no more than $\frac{K}{\epsilon_{1}}$. The uniform $W^{1,2}$ bounded of $(\nabla_{i},u_{i})$ implies there is a subsequence of $\{(\nabla_{i},u_{i})\}$ which weakly converges to a Yang-Mills-Higgs pair $(\nabla_{\infty},u_{\infty})$ in $\Omega^{1}(\mathfrak{g}_{E})\times\Omega^{0}(E)$. And by lemma 5.6, the convergence is smooth in $M\backslash\Sigma$. For any $x_{0}\in\Sigma$, choose $R_{0}<i(M)$ such that $B_{R_{0}}(x_{0})\cap\Sigma=\{x_{0}\}$. Assume $\nabla_{i}=d+A_{i}$ on $B_{R_{0}}(x_{0})$. Define

[TABLE]

Then $\lim\limits_{i\to\infty}r_{i}^{1}=0$. Assume

[TABLE]

where $\rho_{i}(x)=x_{0}+r_{i}^{1}x$ maps $B_{R}(0)$ to $B_{Rr_{i}^{1}}(x_{0})$ for any $R>0$. The pairs $(\tilde{\nabla}_{1,i},\tilde{u}_{1,i})$ satisfy

[TABLE]

For any $\delta<R_{0}$ and $R>0$, $i$ is large enough such that $\delta>4Rr_{i}^{1}$. Note that $\|R^{\tilde{\triangledown}_{1,i}}\|_{L^{\infty}(B_{R}(0))}\leq 1$. Applying theorem 5.3, lemma 5.4 and lemma 5.6, there is a subsequence of $(\tilde{A}_{1,i},\tilde{u}_{1,i})$ converges to $(\tilde{A}_{1,\infty,R},\tilde{u}_{1,\infty,R})$ on $B_{R}(0)$ smoothly under some gauge transformations $g_{i}$. For simplicity, we assume that the subsequence is $(\tilde{A}_{1,i},\tilde{u}_{1,i})$ itself. By choosing $R_{1}<R_{2}<...\to\infty$ and subsequence repeatedly, we may assume $(\tilde{A}_{1,i}^{g_{i}},\tilde{u}_{1,i}^{g_{i}})$ converges to $(\tilde{A}_{1,\infty},\tilde{u}_{1,\infty})$ in $C^{\infty}(\mathbb{R}^{4})$ satisfying

[TABLE]

which implies $\widetilde{\nabla}_{1,\infty}$ is a Yang-Mills connection and $d^{\tilde{\triangledown}_{1,\infty}}\tilde{u}_{1,\infty}=0$. By Uhlenbeck’s removable singularity theorem (see [18], corollary 4.3) , $\tilde{\nabla}_{1,\infty}$ can be extended to a nontrivial Yang-Mills connection on $S^{4}$ under some gauge transformation. Define

[TABLE]

The following lemma gives a sufficient condition for the energy on necks to vanish.

Lemma 5.7**.**

Assume $(\nabla_{i},u_{i})$ are a sequence of Yang-Mills-Higgs pairs and $\lim\limits_{i\to\infty}r_{i}^{1}=0$. There exists $\epsilon$, such that for any $R,\delta>0$, if

[TABLE]

then

[TABLE]

Proof**.**

For simplicity, we assume $x_{0}=0$. For $l\geq-1$, define

[TABLE]

Divide $A_{i}$ into the radius part $A_{i,r}$ and the sphere part $A_{i,\theta}$, that is, $A_{i}=A_{i,r}+A_{i,\theta}$. Recall that by Bochner’s formula, the equation (2) and $\|u_{i}\|_{L^{\infty}}\leq 1$, we have

[TABLE]

Note that

[TABLE]

and let $h$ tends to [math], we have

[TABLE]

Consider $\tilde{A}_{i,l}(x)=2^{-l}A_{i}(2^{-l}x)$, $\tilde{u}_{i,l}(x)=u_{i}(2^{-l}x)$ and assume $\tilde{\nabla}_{i,l}=d+\tilde{A}_{i,l}$. Note that

[TABLE]

and by Harnack’s inequality, we have

[TABLE]

Theorem 2.8 and corollary 2.9 in [18] show that

Lemma 5.8**.**

*Assume $\nabla=d+A$ be a connection over $\mathfrak{U}_{0}$, there exists $\gamma$ such that if $\|R^{\triangledown}\|_{L^{\infty}(\mathfrak{U}_{0})}\leq\gamma$, then under a gauge transformation, we have $\delta^{\triangledown}A=0$ in $\mathfrak{U}_{-1}$, $\delta_{\theta}^{\triangledown}A_{\theta}=0$ on $S_{-1}$ and $S_{0}$, and $\int_{|x|=r}A_{r}dS=0$ for any $1\leq r\leq 2$. Moreover, there is $\|A\|_{L^{\infty}(\mathfrak{U}_{0})}\leq C\|R^{\triangledown}\|_{L^{\infty}(\mathfrak{U}_{0})}$ and $\|A\|_{L^{2}(\mathfrak{U}_{0})}\leq C\|R^{\triangledown}\|_{L^{2}(\mathfrak{U}_{0})}$. *

By choosing $\epsilon$ small enough, we may assume $|R^{\tilde{\nabla}_{i,l}}|\leq\gamma$ on $\mathfrak{U}_{0}$ and hence there exists a gauge transformation $g_{i,l}\in\mathscr{G}(\mathfrak{U}_{l})$, if denote $A_{i,l}=A_{i}^{g_{i,l}}$ and $u_{i,l}=u_{i}^{g_{i,l}}$, we have

[TABLE]

Choosing $\epsilon$ small enough such that $\{\nabla_{i}\}$ satisfy the condition of lemma 5.5, we may assume $\int_{\mathfrak{U}_{l}}|A_{i,l}|^{2}dV\leq 2^{-2l}C\int_{\mathfrak{U}_{l}}|R^{\triangledown_{i,l}}|^{2}dV$. Assume

[TABLE]

By Stokes’ formula, we have

[TABLE]

Let $\hat{A}_{i,l}(x)=2^{-l}A_{i,l}(2^{-l}x)$ and assume $\|R^{\hat{\triangledown}_{i,l}}\|_{L^{\infty}(\mathfrak{U}_{0})}=2^{-2l}\|R^{\triangledown_{i,l}}\|_{L^{\infty}(\mathfrak{U}_{l})}=2^{-2l}\|R^{\triangledown_{i}}\|_{L^{\infty}(\mathfrak{U}_{l})}\leq\gamma$ by choosing $\epsilon$ small enough, where $\gamma$ is the constant in lemma 5.8. Then we have

[TABLE]

Choose $\epsilon$ small enough such that $C\epsilon\leq\frac{1}{2}$, then

[TABLE]

Then we have

[TABLE]

By equation (2) and Hölder’s inequality, we have

[TABLE]

which tends to 0 as $\delta$ tends to 0 since $2^{-l_{1}}\leq 2\delta$. By Fatou’s lemma, we have

[TABLE]

hence

[TABLE]

Since $\|A_{i,l}\|_{L^{\infty}(S_{l})}\leq 2^{-l}C\|R^{\triangledown_{i}}\|_{L^{\infty}(S_{l})}\leq 2^{l}C\epsilon$, we have

[TABLE]

as $\delta\to 0$. According to the same argument for $S_{l_{2}}$, we conclude that

[TABLE]

Since lemma 5.1 and lemma 5.2 shows that Higgs part $|d^{\triangledown_{i}}u_{i}|^{2}+\frac{\lambda}{4}(1-|u_{i}|^{2})^{2}$ has no concentration point, we have

[TABLE]

For any $\delta,R>0$, define $E_{i}=\{r_{i}\in(\frac{Rr_{i}^{1}}{2},2\delta)\mid\mathscr{A}(\nabla_{i},u_{i};B_{2r_{i}}(x_{0})\backslash B_{r_{i}}(x_{0}))\geq\epsilon\}$. If $\cup_{j}\cap_{i>j}E_{i}\neq\varnothing$, define equivalent classes of $\{(r_{1},r_{2},...)\mid r_{i}\in E_{i}\}$ such that $\{r_{i}\}$ equals to $\{r_{i}^{\prime}\}$ if and only if $\liminf\frac{r_{i}}{r_{i}^{\prime}}>0$ and $\limsup\frac{r_{i}}{r_{i}^{\prime}}<\infty$. And define $\{r_{i}\}>\{r_{i}^{\prime}\}$ if and only if $\liminf\frac{r_{i}^{\prime}}{r_{i}}=0$. Similarly to the estimate of the number of element of $\Sigma$, we have the estimate of the number of equivalent class $L\leq\frac{K}{\epsilon}$. Assume the equivalent classes are $\{r_{i}^{1}\}<\{r_{i}^{2}\}<...<\{r_{i}^{L}\}$. Note that

[TABLE]

For $j=1,...,l$, define

[TABLE]

Since $\mathscr{A}(\nabla_{i},u_{i};B_{\delta r_{i}^{j}}(x_{0})\backslash B_{Rr_{i}^{j-1}}(x_{0}))=\mathscr{A}(\widetilde{\nabla}_{j,i},\tilde{u}_{j,i};B_{\delta}(0)\backslash B_{R{r_{i}^{j-1}}/{r_{i}^{j}}}(0))$, where $\lim\limits_{i\to\infty}\frac{r_{i}^{j-1}}{r_{i}^{j}}=0$ and for any $r\in(\frac{Rr_{i}^{j-1}}{2r_{i}^{j}},2\delta)$, we have

[TABLE]

(otherwise there exists an equivalent class between $\{r_{i}^{j-1}\}$ and $\{r_{i}^{j}\}$). By lemma 5.7, we have

[TABLE]

Similarly, $\{r_{i}^{L}\}$ is the largest equivalent implies $\{(\nabla_{i},u_{i})\}$ satisfy (31) by replacing $\{r_{i}^{1}\}$ with $\{r_{i}^{L}\}$ and thus

[TABLE]

The uniformly $W^{1,2}$ bound of $\widetilde{\nabla}_{j,i}$ implies that there is a weakly limit $\widetilde{\nabla}_{j,\infty}$ in $\mathbb{R}^{4}-\{0\}$ and the removable singularity theorem shows that it could be extended to a Yang-Mills connection over $S^{4}$. The convergence may be not smooth in $S^{4}$, and we can repeat the bubble-neck decomposition at each blow-up point as above. This process must stop after finite steps by the uniform energy bound. Hence

[TABLE]

where $\widetilde{\nabla}_{j,\infty,k}$ are the bubbles of $\widetilde{\nabla}_{j,\infty}$. For simplicity, we assume $\Sigma=\{x_{0}\}$. Finally, by choosing a subsequence, we have

[TABLE]

Then we finish the prove.

Remark 5.2

If we consider the Higgs fields taking values in $\Omega^{0}(\mathfrak{g}_{E})$, we can get the similar energy identity.

Theorem 5.9**.**

Assume $\{(\nabla_{i},\Phi_{i})\}$ is a family of Yang-Mills-Higgs pairs and $\mathscr{A}(\nabla_{i},\Phi_{i})\leq K$, where $\Phi\in\Omega^{0}(\mathfrak{g}_{E})$. Then there is a finite subset $\Sigma=\{x_{1},...,x_{l}\}\subset M$, a Yang-Mills-Higgs pair $(\nabla_{\infty},\Phi_{\infty})$ on $M\backslash\Sigma$ and Yang-Mills connections $\{\widetilde{\triangledown}_{jk}\mid 1\leq j\leq l,1\leq k\leq K_{j}\}$ over $S^{4}$, such that there is a subsequence of $\{(\nabla_{i},\Phi_{i})\}$ converges to $(\nabla_{\infty},\Phi_{\infty})$ in $C^{\infty}_{loc}(M\backslash\Sigma)$ under gauge transformations and

[TABLE]

Conflict interests

There is no conflict of interest.

Acknowledgment

All authors would like to thank Prof. Jiayu Li for his encouragement and constant help. The first author is supported by National Key R$\&$D Program of China 2022YFA1005400 and NFSC No.12031017 and the second author is supported by NSFC No.12001532.