Morse theory for the Yang-Mills energy function near flat connections

Abstract

A result (Corollary 4.3) in an article by Uhlenbeck (1985) asserts that the $W^{1,p}$-distance between the gauge-equivalence class of a connection $A$ and the moduli subspace of flat connections $M(P)$ on a principal $G$-bundle $P$ over a closed Riemannian manifold $X$ of dimension $d\ge 2$ is bounded by a constant times the $L^p$ norm of the curvature, $\Vert F_A{\Vert }_{L^p(X)}$, when $G$ is a compact Lie group, $F_A$ is $L^p$-small, and $p>d/2$. While we prove that this estimate holds when the Yang-Mills energy function on the space of Sobolev connections is Morse-Bott along the moduli subspace $M(P)$ of flat connections, it does not hold when the Yang-Mills energy function fails to be Morse-Bott, such as at the product connection in the moduli space of flat $\mathrm{SU}(2)$ connections over a real two-dimensional torus. However, we prove that a useful modification of Uhlenbeck's estimate always holds provided one replaces $\Vert F_A{\Vert }_{L^p(X)}$ by a suitable power $\Vert F_A{\Vert }_{L^p(X)}^{\lambda }$, where the positive exponent $\lambda$ reflects the structure of non-regular points in $M(P)$. The proof of our refinement involves gradient flow and Morse theory for the Yang-Mills energy function on the quotient space of Sobolev connections and a Lojasiewicz distance inequality for the Yang-Mills energy function. A special case of our estimate, when $X$ has dimension four and the connection $A$ is anti-self-dual, was proved by Fukaya (1998) by entirely different methods. Lastly, we prove that if $A$ is a smooth Yang-Mills connection with small enough energy, then $A$ is necessarily flat.

98 pages. The article draws on supporting material from our articles arXiv:1706.09349, arXiv:1510.03817, arXiv:1510.03815, arXiv:1502.00668, arXiv:1412.4114, and arXiv:1409.1525. We use one figure from arXiv:1301.0164 due to Hedden, Herald, and Kirk by permission of those authors. Additional detail has been added to Appendix A, where a key example is explained

References in corpus (8)

• Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds
• Lojasiewicz-Simon gradient inequalities for analytic and Morse-Bott functions on Banach spaces
• Discreteness for energies of Yang-Mills connections over four-dimensional manifolds
• Optimal Lojasiewicz-Simon inequalities and Morse-Bott Yang-Mills energy functions
• A proof on energy gap for Yang-Mills connection
• Removability of a codimension four singular set for solutions of a yang mills higgs equation with small energy
• Energy Identity for Stationary Yang Mills
• Conditions of smoothness of moduli spaces of flat connections and of character varieties

Cited by in corpus (1)

• Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds