Morse Theory for the Yang-Mills Energy Function near the Moduli Subspace of Flat Connection

A corollary in a well-known 1985 article by Uhlenbeck 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<= 2 is bounded by a constant times the L^p norm of the curvature, |F_A|_{L^p(X)}, when G is a compact Lie group, F_A is L^p-small, and p>d/2. While true when the Yang-Mills energy function is Morse-Bott along M(P), Uhlenbeck’s estimate does not hold in examples when the Yang--Mills energy function on the quotient space B(P) of Sobolev connections is not Morse--Bott along M(P), such as the moduli space of flat SU(2) connections over a real two-dimensional torus. However, we prove that a useful modification of Uhlenbeck’s estimate does hold provided one replaces |F_A|_{L^p(X)} by a suitable power |F_A|_{L^p(X)}^c, where the positive exponent, c, reflects the structure of the singularities 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. Moreover, our method shows that M(P) is a deformation retract of an open neighborhood in B(P), generalizing a proof by Lojasiewicz of a conjecture of Whitney that the zero set of a real analytic function on Euclidean space is a deformation retract of an open neighborhood. 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, apparently unaware of Uhlenbeck’s prior work. Lastly, we prove that if A is a smooth, stationary Yang-Mills connection with small enough L^2 energy, then A is necessarily flat.