A NEW HIGHER ORDER YANG–MILLS–HIGGS FLOW ON RIEMANNIAN $4$-MANIFOLDS

Abstract

Let $(M,g)$ be a closed Riemannian $4$ -manifold and let E be a vector bundle over M with structure group G, where G is a compact Lie group. We consider a new higher order Yang–Mills–Higgs functional, in which the Higgs field is a section of $\Omega ^0(\text {ad}E)$ . We show that, under suitable conditions, solutions to the gradient flow do not hit any finite time singularities. In the case that E is a line bundle, we are able to use a different blow-up procedure and obtain an improvement of the long-time result of Zhang [‘Gradient flows of higher order Yang–Mills–Higgs functionals’, J. Aust. Math. Soc. 113 (2022), 257–287]. The proof relies on properties of the Green function, which is very different from the previous techniques.

1 Introduction

Let $(M,g)$ be a closed Riemannian manifold of real dimension $4$ and let E be a vector bundle over M with structure group G, where G is a compact Lie group. The Yang–Mills functional, defined on the space of connections of E, is given by

$$ \begin{align*} \mathcal{YM}(\nabla)=\frac{1}{2}\int_M|F_{\nabla}|^2\,d\mathrm{vol}_g, \end{align*} $$

where $\nabla $ is a metric compatible connection, $F_{\nabla }$ denotes the curvature, and the pointwise norm $|\cdot |$ is given by g and the Killing form of $\mathrm {Lie}(G)$ . The connection $\nabla $ is called a Yang–Mills connection of E if it satisfies the Yang–Mills equation:

$$ \begin{align*} D^*_{\nabla}F_{\nabla}=0. \end{align*} $$

A solution of the Yang–Mills flow is given by a family of connections $\nabla _t:=\nabla (x,t)$ such that

$$ \begin{align*} \frac{\partial \nabla_t}{\partial t}=-D^*_{\nabla_t}F_{\nabla_t}\quad \mbox{in } M\times [0,T). \end{align*} $$

The Yang–Mills flow was initially studied by Atiyah–Bott [2] to understand the topology of the space of connections by infinite dimensional Morse theory.

We consider the Yang–Mills–Higgs k-functional (or Yang–Mills–Higgs k-energy):

(1.1) $$ \begin{align} \mathcal{YMH}_k(\nabla,u)=\frac{1}{2}\int_M[|\nabla^{(k)}F_{\nabla}|^2+|\nabla^{(k+1)}u|^2]\,d\mathrm{vol}_g, \end{align} $$

where $k\in \mathbb {N}\cup \{0\}$ , $\nabla $ is a connection on E and u is a section of $\Omega ^0(\text {ad}E)$ . In [13], we considered the case when u is a section of $\Omega ^0(E)$ . When $k=0$ , (1.1) is the Yang–Mills–Higgs functional studied in [4, 5]. In [4], Hassell proved the global existence of the Yang–Mills–Higgs flow in 3-dimensional Euclidean space. In [5], Hong–Tian studied the global existence of the Yang–Mills–Higgs flow in 3-dimensional hyperbolic space. Their results yield non-self-dual Yang–Mills connections on $S^4$ . The Yang–Mills–Higgs flow has aroused much attention. For example, Li–Zhang [8] and Song–Wang [10] studied the asymptotic behaviour at time infinity of some Yang–Mills–Higgs flows.

The Yang–Mills–Higgs k-system, that is, the corresponding Euler–Lagrange equations of (1.1), is

$$ \begin{align*} \begin{cases} \displaystyle(-1)^kD^*_{\nabla}\Delta_{\nabla}^{(k)}F_{\nabla}+\sum_{v=0}^{2k-1}P_1^{(v)}[F_{\nabla}]+P_2^{(2k-1)}[F_{\nabla}] +\sum_{i=0}^k\nabla^{*(i)}(\nabla^{(k+1)}u\ast \nabla^{(k-i)}u)=0,\\ \nabla^{*(k+1)}\nabla^{(k+1)}u=0, \end{cases} \end{align*} $$

where $\Delta _{\nabla }^{(k)}$ denotes k iterations of the Bochner Laplacian $-\nabla ^*\nabla $ and the notation P is defined in (2.1). A solution of the Yang–Mills–Higgs k-flow is given by a family of pairs $(\nabla (x,t),u(x,t)):=(\nabla _t,u_t)$ such that

(1.2) $$ \begin{align} \begin{cases} \displaystyle\dfrac{\partial \nabla_t}{\partial t}&\hspace{-9pt}\displaystyle=(-1)^{(k+1)}D^*_{\nabla_t}\Delta_{\nabla_t}^{(k)}F_{\nabla_t}+\sum\limits_{v=0}^{2k-1}P_1^{(v)}[F_{\nabla_t}] \\ &\displaystyle +\ P_2^{(2k-1)}[F_{\nabla_t}]+\sum\limits_{i=0}^k\nabla_t^{*(i)}(\nabla_t^{(k+1)}u_t\ast \nabla_t^{(k-i)}u_t), \\ \dfrac{\partial u_t}{\partial t}&\hspace{-9pt}=-\ \nabla^{*(k+1)}_t\nabla_t^{(k+1)}u_t, \quad \mbox{in } M\times [0,T). \end{cases} \end{align} $$

When $k=0$ , the flow (1.2) is a Yang–Mills–Higgs flow [5].

From an analytic point of view, the Yang–Mills–Higgs k-flow (1.2) admits similar properties to the case in which the Higgs field takes values in $\Omega ^0(E)$ . In fact, by the approach in [13], we can prove the following theorem.

Theorem 1.1. Let E be a vector bundle over a closed Riemannian $4$ -manifold $(M,g)$ and k be an integer with $k> 1$ . For every smooth initial data $(\nabla _0,u_0)$ , there exists a unique smooth solution $(\nabla _t,u_t)$ to the Yang–Mills–Higgs k-flow (1.2) in $M\times [0,+\infty )$ .

Our motivation for considering such flows comes from recent work of Waldron who proved long-time existence for the Yang–Mills flow [12], thereby settling a long standing conjecture. In the context of the Yang–Mills–Higgs flow, it is still unknown whether the flow exists for all times on a Riemannian 4-manifold. The above theorem shows that provided $k> 1$ , the Yang–Mills–Higgs k flow does obey long time existence on a 4-manifold. A question that arises at this point is to understand what is the optimum value for k. By assuming our bundle E is a line bundle, we are able to make progress on this question and show that long-time existence holds for all positive k.

Theorem 1.2. Let E be a line bundle over a closed Riemannian $4$ -manifold $(M,g)$ and k be an integer with $k>0$ . For every smooth initial data $(\nabla _0,u_0)$ , there exists a unique smooth solution $(\nabla _t,u_t)$ to the Yang–Mills–Higgs k-flow (1.2) in $M\times [0,+\infty )$ .

At present, we do not know if this theorem is optimal, meaning that we cannot rule out long-time existence occurring for $k=0$ .

The proof of Theorem 1.1 involves local $L^2$ derivative estimates, energy estimates and blow-up analysis. An interesting aspect of this work is that by using a different blow-up procedure, we are able to obtain a proof of Theorem 1.2, which may be of independent interest. Another interesting aspect is that the proof of long-time existence obstruction (see Theorem 3.7) relies on properties of the Green function, which is very different from the previous techniques in [6, 9, 13].

2 Preliminaries

In this section, we introduce the basic setup and notation that will be used throughout the paper. We follow the notation of [6, 9, 13].

Let E be a vector bundle over a smooth closed manifold $(M,g)$ of real dimension n. The set of all smooth unitary connections on E will be denoted by $\mathcal {A}_E$ . A given connection $\nabla \in \mathcal {A}_E$ can be extended to other tensor bundles by coupling with the corresponding Levi–Civita connection $\nabla _M$ on $(M,g)$ .

Let $D_{\nabla }$ be the exterior derivative, or skew symmetrisation of $\nabla $ . The curvature tensor of E is denoted by

$$ \begin{align*}F_{\nabla}=D_{\nabla}\circ D_{\nabla}.\end{align*} $$

We set $\nabla ^{*},D_{\nabla }^{*}$ to be the formal $L^2$ -adjoints of $\nabla ,D_{\nabla }$ , respectively. The Bochner and Hodge Laplacians are given respectively by

$$ \begin{align*}\Delta_{\nabla}=-\nabla^{*}\nabla, \quad \Delta_{D_{\nabla}}=D_{\nabla}D^*_{\nabla}+D^*_{\nabla}D_{\nabla}.\end{align*} $$

Let $\xi ,\eta $ be p-forms valued in E or $\mathrm {End}(E)$ . Let $\xi \ast \eta $ denote any multilinear form obtained from a tensor product $\xi \otimes \eta $ in a universal way. That is to say, $\xi \ast \eta $ is obtained by starting with $\xi \otimes \eta $ , taking any linear combination of this tensor, taking any number of metric contractions and switching any number of factors in the product. We then have

$$ \begin{align*}|\xi\ast\eta|\leq C|\xi||\eta|.\end{align*} $$

Denote

$$ \begin{align*}\nabla^{(i)}=\underbrace{\nabla\cdots\nabla}_{i\ \mathrm{times}}.\end{align*} $$

We will also use the P notation, as introduced in [7]. Given a tensor $\xi $ , we denote

(2.1) $$ \begin{align} P_v^{(k)}[\xi]:=\sum_{w_1+\cdots+w_v=k}(\nabla^{(w_1)}\xi)\ast\cdots\ast(\nabla^{(w_v)}\xi)\ast T, \end{align} $$

where $k,v\in \mathbb {N}$ and T is a generic background tensor dependent only on g.

3 Long-time existence obstruction

We can use De Turck’s trick to establish the local existence of the Yang–Mills–Higgs k-flow. We refer to [6, 9, 13] for more details. As the proof is standard, we will omit the details.

Theorem 3.1 (Local existence)

Let E be a vector bundle over a closed Riemannian manifold $(M,g)$ . There exists a unique smooth solution $(\nabla _t,u_t)$ to the Yang–Mills–Higgs k-flow (1.2) in $M\times [0,\epsilon )$ with smooth initial value $(\nabla _0,u_0)$ .

Following [6, 9], we can derive estimates of Bernstein–Bando–Shi type, similar to [13, Proposition 4.10].

Proposition 3.2. Let $q\in \mathbb {N}$ , $\gamma \in C^{\infty }_c(M)\ (0\leq \gamma \leq 1)$ and $(\nabla _t,u_t)$ be a solution to the Yang–Mills–Higgs k-flow (1.2) defined on $M\times I$ . Suppose $Q=\max \{1,\sup _{t\in I} |F_{\nabla _t}|\}$ , $K=\max \{1,\sup _{t\in I}|u_t|\}$ and $s\geq (k+1)(q+1)$ . For $t\in [0,T)\subset I$ with $T<{1}/{(QK)^{4}}$ , there exists a positive constant $C_q:=C_q(\mathrm {dim} (M),\mathrm {rk} (E),G,q,k,s,g,\gamma )\in \mathbb {R}_{>0}$ such that

$$ \begin{align*} \|\gamma^{s}\nabla_t^{(q)}F_{\nabla_t}\|_{L^2}^2+\|\gamma^{s}\nabla_t^{(q)}u_t\|_{L^2}^2 \leq C_qt^{-{q}/{(k+1)}}\sup_{t\in[0,T)}(\|F_{\nabla_t}\|^2_{L^2}+\|u_t\|^2_{L^2}). \end{align*} $$

The following corollary is a direct consequence of the above proposition and will be used in the blow-up analysis. The proof relies on the Sobolev embedding, $W^{p,2}\subset C^0$ provided $p>{n}{/2}$ , and Kato’s inequality $|d|u_t||\leq |\nabla _tu_t|$ . More details can be found in Kelleher’s paper [6, Corollary 3.14].

Corollary 3.3. Suppose $(\nabla _t,u_t)$ solves the Yang–Mills–Higgs k-flow (1.2) defined on $M\times [0,\tau ]$ . Set $\bar {\tau }:=\min \{\tau ,1\}$ . Suppose $Q=\max \{1,\sup _{t\in [0,\bar {\tau }]} |F_{\nabla _t}|\}$ , $K\,{=}\,\max \{1,\sup _{t\in [0,\bar {\tau }]}|u_t|\}$ . Assume $\gamma \in C^{\infty }_c(M)\ (0\leq \gamma \leq 1)$ . For $s,l\in \mathbb {N}$ with $s\geq (k+1)(l+1)$ , there exists $C_l:=C_l(\dim (M),\mathrm {rk}(E),K,Q,G,s,k,l,\tau ,g,\gamma )\in \mathbb {R}_{>0}$ such that

$$ \begin{align*} \sup_M(|\gamma^{s}\nabla_{\bar{\tau}}^{(l)}F_{\nabla_{\bar{\tau}}}|^2+|\gamma^{s}\nabla_{\bar{\tau}}^{(l)}u_{\bar{\tau}}|^2) \leq C_l\sup_{M\times [0,{\bar{\tau}})}(\|F_{\nabla_t}\|^2_{L^2}+\|u_t\|^2_{L^2}). \end{align*} $$

From Corollary 3.3, we deduce the following corollary, which can be used for finding obstructions to long-time existence.

Corollary 3.4. Suppose $(\nabla _t,u_t)$ solves the Yang–Mills–Higgs k-flow (1.2) defined on $M\times [0,T)$ for $T\in [0,+\infty )$ . Suppose

$$ \begin{align*}Q=\max\{1,\sup\limits_{t\in [0,T)} |F_{\nabla_t}|,\sup\limits_{t\in [0,T)} \|F_{\nabla_t}\|_{L^2}\}\end{align*} $$

and

$$ \begin{align*} K=\max\{1,\sup\limits_{t\in [0,T)}|u_t|,\sup\limits_{t\in [0,T)}\|u_t\|_{L^2}\}\end{align*} $$

are finite. Assume $\gamma \in C^{\infty }_c(M)\ (0\leq \gamma \leq 1)$ . Then, for $t\in [0,T)$ and $s,l\in \mathbb {N}$ with ${s\geq (k+1)(l+1)}$ , there exists $C_l:=C_l(\nabla _0,u_0,\dim (M),\mathrm {rk}(E),K,Q,G,s,k,l,g,\gamma )\in \mathbb {R}_{>0}$ such that

$$ \begin{align*} \sup_{M\times[0,T)}(|\gamma^{s}\nabla_t^{(l)}F_{\nabla_{t}}|^2+|\gamma^{s}\nabla_t^{(l)}u_t|^2)\leq C_l. \end{align*} $$

We will use Corollary 3.4 to show that the only obstruction to long-time existence of the Yang–Mills–Higgs k-flow (1.2) is a lack of a supremal bound on $|F_{\nabla _t}|+|\nabla _tu_t|$ . Before doing so, we need the following proposition, which is similar to [13, Proposition 4.15].

Proposition 3.5. Suppose $(\nabla _t,u_t)$ is a solution to the Yang–Mills–Higgs k-flow (1.2) defined on $M\times [0,T)$ for $T\in [0,+\infty )$ . Suppose that for all $l\in \mathbb {N}\cup \{0\}$ , there exists $C_l\in \mathbb {R}_{>0}$ such that

$$ \begin{align*}\max\bigg\{\sup_{M\times[0,T)}\bigg|\nabla_t^{(l)}\bigg[\frac{\partial \nabla_t}{\partial t}\bigg]\bigg|, \sup_{M\times[0,T)}\bigg|\nabla_t^{(l)}\bigg[\frac{\partial u_t}{\partial t}\bigg]\bigg|\bigg\}\leq C_l.\end{align*} $$

Then $\lim _{t\rightarrow T}(\nabla _t,u_t)=(\nabla _T,u_T)$ exists and is smooth.

The following proposition is straightforward.

Proposition 3.6. Suppose $(\nabla _t,u_t)$ is a solution to the Yang–Mills–Higgs k-flow (1.2) defined on $M\times [0,T)$ . We have

$$ \begin{align*} \sup_{t\in[0,T)}\|u_t\|_{L^2}<+\infty. \end{align*} $$

Using Propositions 3.5 and 3.6, we are ready to prove the main result in this section.

Theorem 3.7. Assume E is a line bundle. Suppose $(\nabla _t,u_t)$ is a solution to the Yang–Mills–Higgs k-flow (1.2) for some maximal $T<+\infty $ . Then,

$$ \begin{align*}\sup_{M\times[0,T)}(|F_{\nabla_t}|+|\nabla_t u_t|)=+\infty.\end{align*} $$

Proof. Suppose to the contrary that

$$ \begin{align*}\sup_{M\times[0,T)}(|F_{\nabla_t}|+| \nabla_t u_t|)<+\infty,\end{align*} $$

which means that

$$ \begin{align*}\sup_{M\times[0,T)}|F_{\nabla_t}|<+\infty, \quad \sup_{M\times[0,T)}| \nabla_t u_t|<+\infty.\end{align*} $$

Denote by $G_t(x,y)$ the Green function associated to the operator $\Delta _{\nabla _t}$ . Then for any fixed $x\in M$ , $\|\nabla _0 G_t(x,\cdot )\|_{L^{\infty }(M)}\leq C_G$ for a constant $C_G$ from [1, Appendix A]. Note that $\nabla _tG_t-\nabla _0 G_t=[\nabla _t-\nabla _0,G_t]=0$ . We conclude that $\|\nabla _t G_t\|_{L^{\infty }(M)}$ is also uniformly bounded. Therefore, using the properties of the Green function in [1, Appendix A],

$$ \begin{align*} \begin{split} \bigg|u_t(x)-\frac{1}{\text{Vol}(M)}\int_M u_t(y)\,{d}y\bigg|&=\bigg|\int_M \Delta_{\nabla_t} G_t(x,y) u_t(y)\,{d}y\bigg|\\ &=\bigg|\int_M \nabla_t G_t(x,y)\nabla_tu_t(y)\,{d}y\bigg|\\ &< +\infty, \end{split} \end{align*} $$

which together with Proposition 3.6 implies

$$ \begin{align*}\sup_{M\times[0,T)}|u_t|<+\infty.\end{align*} $$

For all $t\in [0,T)$ and $l\in \mathbb {N}\cup \{0\}$ , by Corollary 3.4, $\sup _{M}(|\nabla _t^{(l)}F_{\nabla _{t}}|^2+|\nabla _t^{(l)}u_t|^2)$ is uniformly bounded and so by Proposition 3.5, $\lim _{t\rightarrow T}(\nabla _t,u_t)=(\nabla _T,u_T)$ exists and is smooth. However, by local existence (Theorem 3.1), there exists $\epsilon>0$ such that $(\nabla _t,u_t)$ exists over the extended domain $[0,T+\epsilon )$ , which contradicts the assumption that T is maximal. Thus, we prove the theorem.

4 Blow-up analysis

In this section, we will address the possibility that the Yang–Mills–Higgs k-flow admits a singularity given no bound on $|F_{\nabla _t}|+|\nabla _tu_t|$ . To begin with, we first establish some preliminary scaling laws for the Yang–Mills–Higgs k-flow.

Proposition 4.1. Suppose $(\nabla _t,u_t)$ is a solution to the Yang–Mills–Higgs k-flow (1.2) defined on $M\times [0,T)$ . Define the 1-parameter family $\nabla _t^{\rho }$ with local coefficient matrices given by

$$ \begin{align*}\Gamma_t^{\rho}(x):=\rho \Gamma_{\rho^{2(k+1)}t}(\,\rho x),\end{align*} $$

where $\Gamma _t(x)$ is the local coefficient matrix of $\nabla _t$ . Define the $\rho $ -scaled Higgs field $u_t^{\rho }$ by

$$ \begin{align*}u_t^{\rho}(x):=\rho u_{\rho^{2(k+1)}t}(\,\rho x).\end{align*} $$

Then $(\nabla _t^{\rho },u_t^{\rho })$ is also a solution to the Yang–Mills–Higgs k-flow (1.2) defined on $[0,{T}/{\rho ^{2(k+1)}})$ .

Next we will show that if the curvature coupled with a Higgs field is blowing up as one approaches the maximal time, then one can extract a blow-up limit. The proof will closely follow the arguments in [6, Proposition 3.25] and [13, Theorem 5.2].

Theorem 4.2. Assume E is a line bundle. Suppose $(\nabla _t,u_t)$ is a solution to the Yang–Mills–Higgs k-flow (1.2) defined on some maximal time interval $[0,T)$ with $T< +\infty $ . Then there exists a blow-up sequence $(\nabla ^i_t,u^i_t)$ which converges pointwise to a smooth solution $(\nabla ^{\infty }_t,u^{\infty }_t)$ to the Yang–Mills–Higgs k-flow (1.2) defined on the domain $\mathbb {R}^n\times \mathbb {R}_{<0}$ .

Proof. From Theorem 3.7,

$$ \begin{align*}\lim_{t\rightarrow T}\sup_M(|F_{\nabla_t}|+|\nabla_t u_t|)=+\infty.\end{align*} $$

Therefore, we can choose a sequence of times $t_i\nearrow T$ within $[0,T)$ and a sequence of points $x_i$ , such that

$$ \begin{align*}|F_{\nabla_{t_i}}(x_i)|+|\nabla_{t_i} u_{t_i}(x_i)| =\sup_{M\times [0,t_i]}(|F_{\nabla_t}|+|\nabla_t u_t|).\end{align*} $$

Let $\{\rho _i\}\subset \mathbb {R}_{>0}$ be constants to be determined. Define $\nabla _t^i(x)$ by

$$ \begin{align*}\Gamma_t^i(x)=\rho_i^{{1}/{2(k+1)}}\Gamma_{\rho_i t+t_i}(\,\rho_i^{{1}/{2(k+1)}} x+x_i)\end{align*} $$

and

$$ \begin{align*}u_t^i(x)=\rho_i^{{1}/{2(k+1)}}u_{\rho_i t+t_i}(\,\rho_i^{{1}/{2(k+1)}} x+x_i).\end{align*} $$

By Proposition 4.1, $(\nabla ^i_t,u^i_t)$ are also solutions to the Yang–Mills–Higgs k-flow (1.2) and the domain for each $(\nabla ^i_t,u^i_t)$ is $B_0(\,\rho _i^{-{1}/{2(k+1)}})\times [-{t_i}/{\rho _i},{(T-t_i)}/{\rho _i})$ . We observe that

$$ \begin{align*}F_t^i(x):=F_{\nabla_t^i}(x)=\rho_i^{{1}/{(k+1)}}F_{\nabla_{\rho_i t+t_i}}(\,\rho_i^{{1}/{2(k+1)}}x+x_i),\end{align*} $$

which means that

$$ \begin{align*} \begin{split} &\sup_{t\in[-{t_i}/{\rho_i},{T-t_i}/{\rho_i})} (|F_t^i(x)|+|\nabla_{t}^iu_t^i(x)|)\\ &\qquad =\rho_i^{{1}/{(k+1)}}\sup_{t\in[-{t_i}/{\rho_i},{T-t_i}/{\rho_i})}(|F_{\nabla_{\rho_i t+t_i}}(\,\rho_i^{{1}/{2(k+1)}} x+x_i)|+|\nabla_{\rho_i t+t_i} u_{\rho_i t+t_i}(\,\rho_i^{{1}/{2(k+1)}} x+x_i)|)\\ &\qquad =\rho_i^{{1}/{(k+1)}}\sup_{t\in[0,t_i]}(|F_{\nabla_t}(x)|+|\nabla_tu_t(x)|)\\ &\qquad =\rho_i^{{1}/{(k+1)}}(|F_{\nabla_{t_i}}(x_i)|+|\nabla_{t_i}u_{t_i}(x_i)|). \end{split} \end{align*} $$

Therefore, setting

$$ \begin{align*}\rho_i=(|F_{\nabla_{t_i}}(x_i)|+|\nabla_{t_i}u_{t_i}(x_i)|)^{-(k+1)}\end{align*} $$

gives

(4.1) $$ \begin{align} 1=|F^i_0(0)|+|\nabla_0^i u_0^i(0)|=\sup_{t\in[-{t_i}/{\rho_i},0]}(|F_t^i(x)|+|\nabla_t^i u_t^i(x)|). \end{align} $$

Now, we are ready to construct smoothing estimates for the sequence $(\nabla ^i_t,u^i_t)$ . Let $y\in \mathbb {R}^n$ , $\tau \in \mathbb {R}_{\leq 0}$ . For any $s\in \mathbb {N}$ ,

$$ \begin{align*}\sup_{t\in [\tau-1,\tau]}(|\gamma^s_y F^i_t(x)|+|\gamma^s_y \nabla_t^i u^i_t(x)|)\leq 1.\end{align*} $$

Note that E is a line bundle and, similar to the proof of Theorem 3.7, it suffices to use Corollary 3.3. Then for all $q\in \mathbb {N}$ , one may choose $s\geq (k+1)(q+1)$ so that there exists a positive constant $C_q$ such that

$$ \begin{align*} \begin{split} &\sup_{x\in B_y({1}/{2})}(|(\nabla^i_{\tau})^{(q)} F^i_{\tau}(x)|+|(\nabla^i_{\tau})^{(q)} u^i_{\tau}(x)|)\\ &\quad \leq\sup_{x\in B_y(1)}(|\gamma^s_y(\nabla^i_{\tau})^{(q)} F^i_{\tau}(x)|+|\gamma^s_y(\nabla^i_{\tau})^{(q)} u^i_{\tau}(x)|) \leq C_q. \end{split} \end{align*} $$

Then by the Coulomb gauge theorem of Uhlenbeck [11, Theorem 1.3] (see also [5]) and the gauge patching theorem [3, Corollary 4.4.8], passing to a subsequence (without changing notation) and in an appropriate gauge, $(\nabla ^i_t,u^i_t)\rightarrow (\nabla ^{\infty }_t,u^{\infty }_t)$ in $C^{\infty }$ .

5 Proof of Theorem 1.2

The following energy estimates are similar to the ones in [13, Section 6].

Proposition 5.1. Suppose $(\nabla _t,u_t)$ is a solution to the Yang–Mills–Higgs k-flow (1.2) defined on $M\times [0,T)$ . Then the Yang–Mills–Higgs k-energy (1.1) is decreasing along the flow (1.2).

Proposition 5.2. Suppose $(\nabla _t,u_t)$ is a solution to the Yang–Mills–Higgs k-flow (1.2) defined on $M^4\times [0,T)$ with $T< +\infty $ . Then the Yang–Mills–Higgs energy

$$ \begin{align*} \mathcal{YMH}(\nabla_t,u_t)=\frac{1}{2}\int_M[|F_{\nabla_t}|^2+|\nabla_t u_t|^2]\,d\mathrm{vol}_g \end{align*} $$

is bounded along the flow (1.2).

Next, we will complete the proof of Theorem 1.2. To accomplish this, we first show that the $L^p$ -norm controls the $L^{\infty }$ -norm by blow-up analysis.

Proposition 5.3. Assume E is a line bundle. Suppose $(\nabla _t,u_t)$ is a solution to the Yang–Mills–Higgs k-flow (1.2) defined on $M^4\times [0,T)$ and

$$ \begin{align*}\sup_{t\in[0,T)}(\|F_{\nabla_t}\|_{L^p}+\|\nabla_t u_t\|_{L^p})<+\infty.\end{align*} $$

If $p>2$ , then

$$ \begin{align*}\sup_{t\in[0,T)}(\|F_{\nabla_t}\|_{L^{\infty}}+\|\nabla_t u_t\|_{L^{\infty}})<+\infty.\end{align*} $$

Proof. To obtain a contradiction, assume

$$ \begin{align*}\sup_{t\in[0,T)}(\|F_{\nabla_t}\|_{L^{\infty}}+\|\nabla_t u_t\|_{L^{\infty}})=+\infty.\end{align*} $$

As we did in Theorem 4.2, we can construct a blow-up sequence $(\nabla ^i_t,u^i_t)$ , with blow-up limit $(\nabla ^{\infty }_t,u^{\infty }_t)$ . Noting (4.1), by Fatou’s lemma and the natural scaling law,

$$ \begin{align*} \begin{split} \|F_{\nabla^{\infty}_t}\|^p_{L^p}+\|\nabla_t^{\infty} u^{\infty}_t \|^p_{L^p} &\leq \lim_{i\rightarrow +\infty}\inf(\|F_{\nabla^{i}_t}\|^p_{L^p}+\|\nabla_t^i u^{i}_t \|^p_{L^p})\\ &\leq \lim_{i\rightarrow +\infty}\rho_i^{{(2p-4)}/{(2k+2)}}(\|F_{\nabla_t}\|^p_{L^p}+\|\nabla_t u_t\|^p_{L^p}). \end{split} \end{align*} $$

Since $\lim _{i\rightarrow +\infty }\rho _i^{{(2p-4)}/{(2k+2)}}=0$ when $p>2$ , the right-hand side of the above inequality tends to zero, which is a contradiction since the blow-up limit has nonvanishing curvature.

Now we are ready to give the proof of Theorem 1.2.

Proof of Theorem 1.2

By the Sobolev embedding theorem, we can solve for p such that $W^{k,2}\subset L^{p}$ , when $k>0$ . Using the interpolation inequalities [7, Corollary 5.5]:

$$ \begin{align*} \begin{split} \|F_{\nabla_t}\|_{L^{p}}+\|\nabla_t u_t\|_{L^{p}} & \leq CS_{k,p}\sum_{j=0}^k(\|\nabla_t^{(j)}F_{\nabla_t}\|^2_{L^2}+\|\nabla_t^{(j)}u_t\|^2_{L^2}+1)\\ & \leq CS_{k,p}(\|\nabla_t^{(k)}F_{\nabla_t}\|^2_{L^2}+\|F_{\nabla_t}\|^2_{L^2}+\|\nabla_t^{(k+1)}u_t\|^2_{L^2}+\|u_t\|^2_{L^2}+1)\\ & \leq CS_{k,p}(\mathcal{YMH}_k(\nabla_t,u_t)+\mathcal{YMH}(\nabla_t,u_t)+\|u_t\|^2_{L^2}+1), \end{split} \end{align*} $$

then using Propositions 5.1, 3.6 and 5.2, we conclude that the flow exists smoothly for all time.