A proof on energy gap for Yang-Mills connection

TL;DR

This paper establishes an energy gap result for Yang-Mills connections on compact manifolds, demonstrating a lower bound on the energy without relying on the Lojasiewicz-Simon inequality.

Contribution

It provides a new proof of the energy gap for Yang-Mills connections that avoids the use of the Lojasiewicz-Simon gradient inequality.

Findings

Proves an ${L^{rac{n}{2}}}$-energy gap for Yang-Mills connections.

Provides a novel proof technique avoiding the Lojasiewicz-Simon inequality.

Enhances understanding of the energy landscape of Yang-Mills connections.

Abstract

In this note, we prove an ${L^{\frac{n}{2}}}$-energy gap result for Yang-Mills connections on a principal $G$-bundle over a compact manifold without using Lojasiewicz-Simon gradient inequality (arXiv:1502.00668).

Equations

$$YM(A)={\int }_X|F_A{|}^{2}dvo{l}_g.$$

$$d_A^{\ast }{F}_A=0$$

$${\int }_X|F_A{|}^{\frac{n}{2}}dvo{l}_g\ge C_0$$

$$\|u\|_{L^{p}_{k,A}(X)}:=\big{(}\sum_{j=0}^{k}\int_{X}|\nabla^{j}_{A}u|^{p}dvol_{g}\big{)}^{1/p},$$

$$\Vert u{\Vert }_{L_{k,A}^{\infty }(X)}:=\sum _{j=0}^{k}ess\underset{X}{\sup }|{\nabla }_A^{j}u|.$$

$$\Vert F_A{\Vert }_{L^p(X)}\le \epsilon ,$$

$$\Vert F_A{\Vert }_{L^{\frac{n}{2}}(X)}\le \epsilon ,$$

$$\Vert F_A{\Vert }_{L^{\infty }(X)}\le C\Vert F_A{\Vert }_{L^2(X)}.$$

$$\Vert F_A{\Vert }_{L^p(X)}\le C\Vert F_A{\Vert }_{L^{\infty }(X)}\le C\Vert F_A{\Vert }_{L^2(X)},$$

$$\Vert F_A{\Vert }_{L^2(X)}\le C\Vert F_A{\Vert }_{L^{\frac{n}{2}}(X)}.$$

$$\begin{array}{rl}\Vert F_A{\Vert }_{L^2(X)}& \le C\Vert F_A{\Vert }_{L^{\frac{n}{2}}(X)}^{\frac{n}{4}}\Vert F_A{\Vert }_{L^{\infty }(X)}^{1-\frac{n}{4}}\\ & \le C\Vert F_A{\Vert }_{\frac{n}{2}(X)}^{\frac{n}{4}}\Vert F_A{\Vert }_{L^2(X)}^{1-\frac{n}{4}}\end{array}$$

$$\Vert F_A{\Vert }_{L^2(X)}\le C\Vert F_A{\Vert }_{L^{\frac{n}{2}}(X)}.$$

$$\Vert F_A{\Vert }_{L^p(X)}\le C\Vert F_A{\Vert }_{L^{\frac{n}{2}}(X)},\forall 2p\ge nandn\ge 2.$$

$$\Vert u^{\ast }(A)-\Gamma {\Vert }_{L_1^q(X)}\le C(q)\Vert F_A{\Vert }_{L^q(X)},$$

$$d_{\Gamma }^{\ast }(u^{\ast }(A)-\Gamma )=0.$$

$$F_{\tilde{A}}=d_{\Gamma }a+a\wedge a.$$

$$0=d_{\tilde{A}}^{\ast }{F}_{\tilde{A}}.$$

$$\begin{array}{rl}0& =(d_{\tilde{A}}^{\ast }{F}_{\tilde{A}},a{)}_{L^2(X)}\\ & =(F_{\tilde{A}},d_{\tilde{A}}a{)}_{L^2(X)}\\ & =(F_{\tilde{A}},d_{\Gamma }a+2a\wedge a{)}_{L^2(X)}\\ & =(F_{\tilde{A}},F_{\tilde{A}}+a\wedge a{)}_{L^2(X)}.\end{array}$$

$$\begin{array}{rl}\Vert F_A{\Vert }_{L^2(X)}^2& =\Vert F_{\tilde{A}}{\Vert }_{L^2(X)}^2\\ & =-(F_{\tilde{A}},a\wedge a{)}_{L^2(X)}\\ & \le \Vert F_{\tilde{A}}{\Vert }_{L^2(X)}\Vert a\wedge a{\Vert }_{L^2(X)}\\ & =\Vert F_A{\Vert }_{L^2(X)}\Vert a\wedge a{\Vert }_{L^2(X)}\end{array}$$

$$\begin{array}{rl}\Vert a\wedge a{\Vert }_{L^2(X)}& \le C\Vert a{\Vert }_{L^4(X)}^2\\ & \le C\Vert a{\Vert }_{L^n(X)}^2\\ & \le C\Vert a{\Vert }_{L_1^{\frac{n}{2}}(X)}^2\\ & \le C\Vert F_A{\Vert }_{L^{\frac{n}{2}}(X)}^2\\ & \le C\Vert F_A{\Vert }_{L^{\infty }(X)}^2\\ & \le C\Vert F_A{\Vert }_{L^2(X)}^2.\end{array}$$

$$\begin{array}{rl}\Vert a\wedge a{\Vert }_{L^2(X)}& \le C\Vert a{\Vert }_{L^4(X)}^2\\ & \le C\Vert a{\Vert }_{L_1^2(X)}^2\\ & \le C\Vert F_A{\Vert }_{L^2(X)}^2.\end{array}$$

$$\Vert F_A{\Vert }_{L^2(X)}^2\le C\Vert F_A{\Vert }_{L^2(X)}^3,$$

Full text

††T. Huang:Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China Hefei, Anhui 230026, PR China; e-mail: oula143@mail.ustc.edu.cn††Mathematics Subject Classification (2010): 58E15;81T13

Keywords. Yang-Mills connection, flat connection, energy gap

1 Introduction

Let $X$ be a compact $n$-dimensional Riemannian manifold with smooth Riemannian metric $g$, $P\rightarrow X$ a principal $G$-bundle over $X$, where $G$ is a compact Lie group. We defined the Yang-Mills functional by

[TABLE]

where $A$ is a $C^{\infty}$-connection on $P$ and $F_{A}$ is the curvature of $A$.

A connection $A$ on $P$ is called Yang-Mills connection, if it is a critical point of $YM$ i.e. it obeys the Yang-Mills equation with respect to the metric $g$:

[TABLE]

In [2], Feehan proved an $L^{\frac{n}{2}}$-energy gap result for Yang-Mills connections on principal $G$-bundle $P$ over arbitrary closed smooth Riemannian manifold with dimensional $n\geq 2$ ([2] Theorem 1.1). Feehan applied the Lojasiewicz-Simon gradient inequality ([2] Theorem 3.2) to remove a positive hypothesis on the Riemannian curvature tensors in a previous $L^{\frac{n}{2}}$-energy gap result due to Gerhardt [3] Theorem 1.2.

In this note, we give another way to prove the $L^{\frac{n}{2}}$-energy gap result of Yang-Mills connection without using the Lojasiewicz-Simon gradient inequality.

Theorem 1.1**.**

([2] Theorem 1.1) Let $X$ be a compact Riemannian manifold without boundary of dimension $n\geq 2$ with smooth Riemannan metric $g$, $P$ be a $G$-bundle over $X$. Then any smooth Yang-Mills connection $A$ over $X$ with compact Lie group $G$ is either satisfies

[TABLE]

for a constant $C_{0}>0$ depending only on $X,n,G$ or the connection $A$ is flat.

2 Preliminaries and basic estimates

We shall generally adhere to the now standard gauge-theory conventions and notation of Donaldson and Kronheimer [1] and Feehan [2]. Throughout our article, $G$ denotes a compact Lie group and $P$ a smooth principal $G$-bundle over a compact Riemannnian manifold $X$ of dimension $n\geq 2$ and endowed with Riemannian metric $g$, $\mathfrak{g}_{P}$ denote the adjoint bundle of $P$, endowed with a $G$-invariant inner product and $\Omega^{p}(X,\mathfrak{g}_{P})$ denote the smooth $p$-forms with values in $\mathfrak{g}_{P}$. Given a connection on $P$, we denote by $\nabla_{A}$ the corresponding covariant derivative on $\Omega^{\ast}(X,\mathfrak{g}_{P})$ induced by $A$ and the Levi-Civita connection of $X$. Let $d_{A}$ denote the exterior derivative associated to $\nabla_{A}$.

For $u\in L^{p}(X,\mathfrak{g}_{P})$, where $1\leq p<\infty$ and $k$ is an integer, we denote

[TABLE]

where $\nabla^{j}_{A}:=\nabla_{A}\circ\ldots\circ\nabla_{A}$ (repeated $j$ times for $j\geq 0$). For $p=\infty$, we denote

[TABLE]

At first, we review a key result due to Uhlenbeck for the connections with $L^{p}$-small curvature ($2p>n$) [5] which provides existence a flat connection $\Gamma$ on $P$, a global gauge transformation $u$ of $A$ to Coulomb gauge with respect to $\Gamma$ and a Sobolev norm estimate for the distance between $\Gamma$ and $A$.

Theorem 2.1**.**

([5] Corollary 4.3 and [2] Theorem 5.1) Let $X$ be a closed, smooth manifold of dimension $n\geq 2$ and endowed with a Riemannian metric, $g$, and $G$ be a compact Lie group, and $2p>n$. Then there are constants, $\varepsilon=\varepsilon(n,g,G,p)\in(0,1]$ and $C=C(n,g,G,p)\in[1,\infty)$, with the following significance. Let $A$ be a $L^{p}_{1}$ connection on a principal $G$-bundle $P$ over $X$. If the curvature $F_{A}$ obeying

[TABLE]

*then there exist a flat connection, $\Gamma$, on $P$ and a gauge transformation ${u\in L^{p}_{2}(X)}$ such that

(1) d^{\ast}_{\Gamma}\big{(}{u}^{\ast}(A)-\Gamma\big{)}=0\ on\ X,

(2) ${\|u^{\ast}(A)-\Gamma\|_{L^{p}_{1,\Gamma}}\leq C\|F_{A}\|_{L^{p}(X)}}$ and

(3) ${\|u^{\ast}(A)-\Gamma\|_{L^{\frac{n}{2}}_{1,\Gamma}}\leq C\|F_{A}\|_{L^{\frac{n}{2}}(X)}}$.*

Next, we also review an other key result due to Uhlenbeck concerning an a priori estimate for the curvature of a Yang-Mills connection over a closed Riemannian manifold.

Theorem 2.2**.**

([4] Theorem 3.5 and [2] Corollary 4.6) Let $X$ be a compact manifold of dimension $n\geq 2$ and endowed with a Riemannian metric $g$, let $A$ be a smooth Yang-Mills connection with respect to the metric $g$ on a smooth $G$-bundle $P$ over $X$. Then there exist constants $\varepsilon=\varepsilon(X,n,g)>0$ and $C=C(X,n,g)$ with the following significance. If the curvature $F_{A}$ obeying

[TABLE]

then

[TABLE]

3 Proof Theorem 1.1

For any $p\geq 1$ , the estimate in Theorem 2.2 yields

[TABLE]

for $C=C(g,n)$.

If $n\geq 4$, by using Höler inequality, we have

[TABLE]

If $n=2,3$, the $L^{p}$ interpolation implies that

[TABLE]

and thus

[TABLE]

Therefore, by combining (3.1)–(3.3), we obtain

[TABLE]

Hence, if we suppose $\|F_{A}\|_{L^{\frac{n}{2}}(X)}$ sufficiently small such that $\|F_{A}\|_{L^{q}(X)}$ ($2q>n$ and $n\geq 2$) satisfies the hypothesis of Theorem 2.1, then Theorem 2.1 provides a flat connection $\Gamma$ on $P$, and a gauge transformation $u\in\mathcal{G}_{P}$ and the estimate

[TABLE]

and

[TABLE]

We denote ${\tilde{A}:=u^{\ast}(A)}$ and ${a:=u^{\ast}(A)-\Gamma}$, then the curvature of $\tilde{A}$ is

[TABLE]

The connection $\tilde{A}$ also satisfies Yang-Mills equation

[TABLE]

Hence taking the $L^{2}$-inner product of (3.4) with $a$, we obtain

[TABLE]

Then we get

[TABLE]

here we use the fact $|F_{u^{\ast}(A)}|=|F_{A}|$ since $F_{u^{\ast}(A)}=u\circ F_{A}\circ u^{-1}$.

If $n\geq 4$:

[TABLE]

where we apply the Sobolev embedding $L^{\frac{n}{2}}_{1}\hookrightarrow L^{n}$.

If $n=2,3$,

[TABLE]

where we apply the Sobolev embedding $L^{2}_{1}\hookrightarrow L^{4}$.

By combining the preceding inequalities we have

[TABLE]

We can choose $\|F_{A}\|_{L^{2}(X)}$ sufficiently small to such that $C\|F_{A}\|_{L^{2}(X)}<1$, hence $\|F_{A}\|_{L^{2}(X)}\equiv 0$ and thus $A$ must be a flat connection. Then we complete the proof.

Acknowledgements

I would like to thank Professor Paul Feehan for helpful comments regarding his article [2].  I thank the anonymous referee for a careful reading of my article and helpful comments and corrections. This work is partially supported by Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences at USTC.