The Flow of Gauge Transformations on Riemannian Surface with Boundary

Abstract

We consider the gauge transformations of a metric G-bundle over a compact Riemannian surface with boundary. By employing the heat flow method, the local existence and the long time existence of generalized solution are proved.

Appendix A Proof of Corollary D

In [20], the existence of Coulomb gauge with respect to a “small” connection on a disk plays an important role in the existence of conservation law. It is quite natural to expect “good estimate” under “good gauge,” and then the existence of conservation law can be obtained from a fixed point argument of systems of elliptic partial differential equations.

One can also obtain the above existence of Coulomb gauge by heat flow method. In fact, as a result of the existence of generalized solution of (1.3) when applied to a special case, we will present a flow version proof of the existence (cf. ([23], Theorem 2.1; [5], Proposition 2)).

In what follows, we will take $\Sigma$ to be a small disk (dimension two), not necessarily with Euclidean metric. A connection of vector bundle E over $\Sigma$, where E is with rank m and structure group $\mathrm {SO}(m)$, can be written as $d+\Omega$, where $\Omega \in \mathfrak {so}(m)\otimes T^*\Sigma$ is a 1-form with value in $\mathfrak {so}(m)$, then for any gauge transformation S,

Thus, if one sets $A_0=d$ as a trivial connection and $A=d+\Omega$, the existence of generalized solution of (1.3) implies that there exists some $S\in C^\infty (\Sigma ,\mathrm {SO}(m))$, such that

$$\begin{aligned} 0=\nabla ^*_{d+\tilde{\Omega }}\tilde{\Omega }=\nabla _d^*\tilde{\Omega }+\left\{ \tilde{\Omega },\tilde{\Omega }\right\} =d^*\tilde{\Omega }. \end{aligned}$$

By Poincaré Lemma, there exists some $\xi \in C^\infty (\Sigma , \mathfrak {so}(m))$, such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \nabla ^\perp \xi =S^{-1}dS+S^{-1}\Omega S, &{}\quad x\in \Sigma \\ \xi =0, &{}\quad x\in \partial \Sigma . \end{array}\right. } \end{aligned}$$

(A.1)

The boundary condition holds since the flow implies that on $\partial \Sigma$,

which shows that $\xi$ is a constant along $\partial \Sigma$, and one can make $\xi$ vanish on boundary since it is determined up to a constant.

To show the estimates hold in Corollary D, we need to construct $S_0$ properly. Parameterize as $x=(y,r)$, where r is the distance to $\partial \Sigma$ and y is the parameter of $\partial \Sigma$. Firstly, let us assume that $\Omega \in C^\infty (\Sigma ,\mathfrak {so}(m)\otimes T^*\Sigma )$ and set the initial value as

$$\begin{aligned} S_0(x)=\exp \left\{ (1-\eta )r\Omega _\nu (x)\right\} . \end{aligned}$$

where $\Omega _\nu (x)=-\Omega (r,y)(\frac{\partial }{\partial {r}})$, and $\eta \in C_0^\infty (\Sigma )$, with $\eta =0$ on $\Sigma _\delta$ and $\eta =1$ on $\Sigma {\setminus }\Sigma _{2\delta }$; moreover, one can require $|\nabla \eta |\le 2/\delta$. To see the compatibility at the corner, note that, on the corner $\partial \Sigma \times \left\{ 0\right\}$, $S=S_0=\mathrm{Id}$, where $\mathrm{Id}$ is the constant section over $\partial \Sigma$, which maps each point to the identity of G,

The energy of $S_0$ can be controlled as

$$\begin{aligned} 2\mathcal {E}(S_0) =\int _\Sigma |S_0^{-1}\mathrm{d}S_0+S_0^{-1}\Omega S_0|^2 \le \int _\Sigma (|\mathrm{d}S_0|^2+|\Omega |^2) =\Vert \mathrm{d}S_0\Vert _2^2+\Vert \Omega \Vert _2^2. \end{aligned}$$

We claim that $\mathcal {E}(S_0)$ can be taken as small as wanted. In fact, note that for $x\in \Sigma _{2\delta }$, the derivative of $\exp$ is uniformly bounded with respect to $\delta$, thus

$$\begin{aligned} |dS_0|\le C\left( (1-\eta )|\Omega _\nu |+r\left( |\nabla \eta ||\Omega _\nu |+|d\Omega _\nu |\right) \right) , \end{aligned}$$

and

$$\begin{aligned} |dS_0|^2\le C|\Omega |_{1,2}^2\left( 1+r^2|\nabla \eta |^2+r^2\right) . \end{aligned}$$

Finally,

$$\begin{aligned} \int _\Sigma |dS_0|^2&=\int _{\Sigma _{2\delta }{\setminus }\Sigma _{\delta }}|dS_0|^2 \le C\int _{\Sigma _{2\delta }{\setminus }\Sigma _{\delta }} |\Omega |_{1,2}^2(1+\delta ^2\cdot 1/\delta ^2+\delta ^2), \end{aligned}$$

which tends to 0 as $\delta \rightarrow 0$ by the absolute continuity of integration.

Thus, for any $\varepsilon >0$, one can construct $S_0$, such that $\Vert dS_0\Vert _2\le \varepsilon$. Since the energy is monotonically decreasing along the flow,

$$\begin{aligned} \Vert \tilde{\Omega }\Vert _2^2=2\mathcal {E}(S)\le 2\mathcal {E}(S_0) \le \Vert \Omega \Vert _2^2+\varepsilon ^2. \end{aligned}$$

Moreover, since $\nabla ^\perp \xi =S^{-1}dS+S^{-1}\Omega S$,

$$\begin{aligned} \Vert dS\Vert _2\le \Vert \Omega \Vert _2+\Vert \nabla ^\perp \xi \Vert _2 =\Vert \Omega \Vert _2+\Vert \tilde{\Omega }\Vert _2 \le 2\Vert \Omega \Vert _2+\varepsilon . \end{aligned}$$

By Poincaré inequality, one concludes that

$$\begin{aligned} \Vert dS\Vert _2+\Vert \xi \Vert _{1,2}\le C(m)\Vert \Omega \Vert _2. \end{aligned}$$

To show the required estimate for $\Omega \in L^2(\Sigma ,\mathfrak {so}(m)\otimes T^*\Sigma )$, let us approximate it by $\Omega _k\in C^\infty (\Sigma ,\mathfrak {so}(m)\otimes T^*\Sigma )$ in $L^2$ with $\Vert \Omega _k\Vert _2\le \Vert \Omega \Vert _2$. For $A_0=\mathrm{d}$, , , run the gauge transformation heat flow,

One obtains $S_k\in W^{1,2}(D,\mathrm {SO}(m))\cap C^\infty$, $\xi _k\in W^{1,2}(D,\mathrm {SO}(m))\cap C^\infty$, which solves (A.1), with

$$\begin{aligned} \Vert dS_k\Vert _2+\Vert \xi _k\Vert _{1,2}\le C(m)\Vert \Omega _k\Vert _2. \end{aligned}$$

Therefore, $S_k$, $\xi _k$ are $W^{1,2}$ bounded, the weak compactness implies that $S_k$, $\xi _k$ converge weakly to some $S_\infty$, $\xi _\infty$ in $W^{1,2}$, respectively. Take limits in (A.1), one finds the required gauge. The weakly lower semi-continuity implies the required estimate, and we finish the proof of Corollary D.