Asymptotic behavior of Yang-Mills flow in higher dimensions.

*(English)*Zbl 1030.53032
Gu, Chaohao (ed.) et al., Differential geometry and related topics. Proceedings of the international conference on modern mathematics and the international symposium on differential geometry, Shanghai, China, September 19-23, 2001. In honor of Professor Su Buchin on the centenary of his birth. Singapore: World Scientific. 16-38 (2002).

Let \(P(M^4,G)\) be a principal bundle over a 4-dimensional compact Riemannian manifold \(M^4\) with a compact simple Lie group \(G\) as its structure group. Twenty years ago, K. Uhlenbeck [Commun. Math. Phys. 83, 31-42 (1982; Zbl 0499.58019)] and S. Sedlacek [Commun. Math. Phys. 86, 515-527 (1982; Zbl 0506.53016)] studied the asymptotic behavior of certain sequences of Yang-Mills connections on \(P(M^4,G)\) and proved their so-called weak compactness theorems on the existence of limiting connections. A few years later, in 1988, H. Nakajima [J. Math. Soc. Japan 40, 383-392 (1988; Zbl 0647.53030)] generalized Uhlenbeck’s weak compactness theorem to the case of dimension \(n\geq 4\).

Now, in the paper under review, the authors turn to the problem of describing the asymptotic behavior of a Yang-Mills flow \(\{A_t \}\) with initial connection \(A_0\) on a principal bundle \(P(M^n,G)\) over a compact Riemannian manifold of dimension \(n>4\).

Whereas it is well-known that such Yang-Mills flows do exist in the short-time range, the problem of the long-time existence for Yang-Mills flows (and their asymptotic behavior) appears to be rather complicated.

Based upon the methods developed in an earlier paper by one of the first author and M. Struwe [Math. Z. 201, 83-103 (1989; Zbl 0652.58024)], where the related evolution problem for harmonic flows \(u_t: M^m\to N^n\) was investigated, the authors of the present article provide a complete proof of the following general theorem:

Let \(P(M^n,G)\) be a principal bundle over an \(n\)-dimensional \((n>4)\) compact Riemannian manifold \(M^n\) with compact simple Lie group \(G\), and let \(\{A_t\}\) be a regular Yang-Mills flow with an initial connection \(A_0\) for any time parameter \(t\geq 0\). Then there is a subsequence \(\{A_{t_i}\}\) of the flow \(\{A_t\}\) and a sequence \(\{\sigma_i\}\) of gauge transformations such that the gauge-transformed sequence \(\{\sigma^*_i (A_{t_i})\}_{i\in\mathbb{N}}\) is \(C^\infty\)-convergent to a Yang-Mills connection \(A_\infty\) which is regular off a subset \(\Sigma_\infty\) in \(M^n\), whose Hausdorff measure \(H^{n-4+k} (\Sigma_\infty)\) is zero for any \(k>0\).

The proof of this deep theorem is obtained by utilizing (and combining) several general inequalities, a monotonicity formula, a partial regularity theorem, and estimates for higher gauge-covariant derivatives of curvatures for Yang-Mills flows, all of which are proved in the course of the paper.

For the entire collection see [Zbl 1007.00074].

Now, in the paper under review, the authors turn to the problem of describing the asymptotic behavior of a Yang-Mills flow \(\{A_t \}\) with initial connection \(A_0\) on a principal bundle \(P(M^n,G)\) over a compact Riemannian manifold of dimension \(n>4\).

Whereas it is well-known that such Yang-Mills flows do exist in the short-time range, the problem of the long-time existence for Yang-Mills flows (and their asymptotic behavior) appears to be rather complicated.

Based upon the methods developed in an earlier paper by one of the first author and M. Struwe [Math. Z. 201, 83-103 (1989; Zbl 0652.58024)], where the related evolution problem for harmonic flows \(u_t: M^m\to N^n\) was investigated, the authors of the present article provide a complete proof of the following general theorem:

Let \(P(M^n,G)\) be a principal bundle over an \(n\)-dimensional \((n>4)\) compact Riemannian manifold \(M^n\) with compact simple Lie group \(G\), and let \(\{A_t\}\) be a regular Yang-Mills flow with an initial connection \(A_0\) for any time parameter \(t\geq 0\). Then there is a subsequence \(\{A_{t_i}\}\) of the flow \(\{A_t\}\) and a sequence \(\{\sigma_i\}\) of gauge transformations such that the gauge-transformed sequence \(\{\sigma^*_i (A_{t_i})\}_{i\in\mathbb{N}}\) is \(C^\infty\)-convergent to a Yang-Mills connection \(A_\infty\) which is regular off a subset \(\Sigma_\infty\) in \(M^n\), whose Hausdorff measure \(H^{n-4+k} (\Sigma_\infty)\) is zero for any \(k>0\).

The proof of this deep theorem is obtained by utilizing (and combining) several general inequalities, a monotonicity formula, a partial regularity theorem, and estimates for higher gauge-covariant derivatives of curvatures for Yang-Mills flows, all of which are proved in the course of the paper.

For the entire collection see [Zbl 1007.00074].

Reviewer: Werner Kleinert (Berlin)

##### MSC:

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |

53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |

##### Keywords:

Yang-Mills connection; Yang-Mills flow; Yang-Mills equation; gauge theory; gauge transformation; evolution equations; Bochner inequalities; regularity theorems
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\textit{Y.-M. Chen} et al., in: Differential geometry and related topics. Proceedings of the international conference on modern mathematics and the international symposium on differential geometry, Shanghai, China, September 19--23, 2001. In honor of Professor Su Buchin on the centenary of his birth. Singapore: World Scientific. 16--38 (2002; Zbl 1030.53032)