Convergence of the Yang-Mills flow on ALE gravitational instantons

Problem Statement
This paper addresses the long-time convergence behavior of the Yang-Mills flow (1.1) on noncompact 4-manifolds, specifically ALE (Asymptotically Locally Euclidean) gravitational instantons. While the convergence of the flow is well-understood in the compact case, the noncompact setting presents significant analytic challenges. Previous work by Gustaffson, Nakanishi, and Tsai demonstrated that on $\mathbb{R}^4$, the parabolic gap theorem fails: there exist initial connections arbitrarily close to the standard instanton that do not converge to an instanton under the flow. The authors aim to establish a "parabolic gap theorem" for ALE gravitational instantons, proving that if the initial energy is sufficiently small (specifically, if the self-dual curvature energy is below a specific threshold), the flow converges smoothly to an anti-self-dual (ASD) instanton.

Methodology
The authors employ a combination of elliptic and parabolic analysis techniques adapted to the noncompact, Ricci-flat geometry of ALE manifolds.

1. Elliptic Gap Theorem and Sobolev Constants: The proof relies on a sharp Sobolev inequality for ALE manifolds (Theorem 2.2), derived from the work of Viaclovsky and Balogh-Kristaly. This inequality, with a constant depending on the order of the finite group $\Gamma$ ($|\Gamma|$), allows for the deduction of an elliptic gap theorem (Theorem 2.4): any finite-energy Yang-Mills connection with self-dual curvature energy $\|F^+\|_{L^2}^2 < 8\pi^2/|\Gamma|$ must be anti-self-dual ($F^+ \equiv 0$).

2. Energy Estimates and $\epsilon$-Regularity: The authors utilize basic energy identities (Lemma 3.2) showing the non-increasing nature of the Yang-Mills functional and the conservation of topological charge. They establish an $\epsilon$-regularity theorem (Lemma 3.3) for the flow, ensuring that small $L^2$ norms of the self-dual curvature imply $L^\infty$ bounds. This is crucial for controlling the curvature near potential singularities or at infinity.

3. Blow-up Analysis and Contradiction: To prove quantitative decay rates, the authors use a contradiction argument involving rescaling (blow-up) techniques (Theorem 4.1). Assuming the decay rate fails, they construct a sequence of rescaled solutions. By analyzing the limits of these sequences (Uhlenbeck compactness), they show that the limit must be an ASD instanton on either the ALE manifold itself or on $\mathbb{R}^4/\Gamma$. The elliptic gap theorem then forces the self-dual part of the curvature to vanish, contradicting the assumption that the decay rate failed.

4. Improved Decay via $L^p$ Estimates: For initial data with $F^+ \in L^p$ ($1 < p < 2$), the authors derive improved decay rates (Theorem 1.1b). This involves showing that once the $L^2$ norm of the self-dual curvature becomes small, the $L^p$ norm remains bounded, which in turn accelerates the $L^\infty$ decay rate to $o(t^{-2/p})$.

5. Convergence to ASD Instanton: Finally, the authors combine the decay estimates with parabolic Moser iteration and heat semigroup estimates to prove that the connection $A(t)$ converges in $C^\infty_{loc}$ and $L^q$ to an ASD instanton $A_\infty$ (Theorem 6.1).

Key Results
The main result is Theorem 1.1, which establishes the following for an $SU(r)$-bundle over an ALE gravitational instanton $(M,g)$ asymptotic to $(\mathbb{R}^4 \setminus \{0\})/\Gamma$:

• Threshold Condition: If the initial connection $A_0$ has finite Yang-Mills energy and satisfies the strict inequality $\|F^+_{A_0}\|_{L^2}^2 < 8\pi^2/|\Gamma|$, the flow exists for all time.
• Decay of Self-Dual Curvature:
  • Generally, $\|F^+(t)\|_{L^\infty} = o(1/t)$ and $\|F^+(t)\|_{L^2} \to 0$ as $t \to \infty$.
  • If additionally $\|F^+_{A_0}\|_{L^p} < \infty$ for $1 < p < 2$, the decay improves to $\|F^+(t)\|_{L^\infty} = o(t^{-2/p})$ and $\|F^+(t)\|_{L^p} \to 0$.
• Convergence: Under these conditions, the flow $A(t)$ converges smoothly (in $C^\infty_{loc} \cap L^q$ for appropriate $q$) to an ASD instanton $A_\infty$. Crucially, the topological charge is preserved: $\kappa(A_\infty) = \kappa(A_0)$.

Significance and Claims
The authors position this work as a noncompact extension of their previous "parabolic gap theorem" established for the 4-sphere. The significance of the results is outlined in four specific motivations:

1. Overcoming Noncompact Obstacles: The paper demonstrates that the failure of the parabolic gap on $\mathbb{R}^4$ (where connections close to the instanton may not converge) can be avoided on ALE manifolds by imposing a mild decay hypothesis on the initial self-dual curvature.
2. Generalization of Oh-Tataru: The result generalizes the work of Oh and Tataru on $\mathbb{R}^4$, which was limited to zero topological charge and energy below $8\pi^2$. This paper extends the result to higher topological charges, showing convergence to an ASD instanton of the same charge rather than the flat connection. The authors suggest this may facilitate generalizations of the Threshold Conjecture for hyperbolic Yang-Mills equations.
3. Weaker Hypotheses than Ricci Flow Stability: The convergence is achieved under significantly weaker assumptions on the initial data compared to stability results for Ricci flow on gravitational instantons. Specifically, the authors do not require the initial connection to be close to a specific ASD instanton, only that its self-dual energy is below the gap threshold.
4. Moduli Space Topology: The authors anticipate that these convergence results will be useful for studying the topology of moduli spaces of ASD instantons on ALE gravitational instantons.

The paper explicitly notes that while short-time existence is assumed (citing existing literature), a detailed treatment of short-time existence and well-posedness for the flow on complete manifolds remains an open question to be addressed separately.