[Paper Review] Energy Identity for Stationary Yang Mills

Aaron Naber, Daniele Valtorta|arXiv (Cornell University)|Oct 10, 2016

Geometric Analysis and Curvature Flows2 references3 citations

TL;DR

This paper establishes the energy identity for stationary Yang-Mills connections on Riemannian manifolds by proving that the defect measure $\nu = e(x)\,d\lambda^{n-4}$, arising in weak limits of Yang-Mills sequences, is quantized: the energy density $e(x)$ at almost every $n-4$-rectifiable point equals the sum of energies of bubbles—Yang-Mills connections on $S^4$—formed via blow-up at $x$. The key contribution is an effective $L^1$-bound on the Hessian of curvature, which implies energy quantization without prior assumptions, and a quantitative bubble tree decomposition with uniform estimates.

ABSTRACT

Given a principal bundle $P o M$ over a Riemannian manifold with compact structure group $G$, let us consider a stationary Yang-Mills connection $A$ with energy $\int_M |F_A|^2\le Λ$. If we consider a sequence of such connections $A_i$, then it is understood that up to subsequence we can converge $A_i o A$ to a singular limit connection such that the energy measures converge $|F_{A_i}|^2 dv_g o |F_A|^2dv_g +ν$, where $ν=e(x)dλ^{n-4}$ is the $n-4$ rectifiable defect measure. Our main result is to show, without additional assumptions, that for $n-4$ a.e. point the energy density $e(x)$ may be computed explicitly as the sum of the bubble energies arising from blow ups at $x$. Each of these bubbles may be realized as a Yang Mills connection over $S^4$ itself. Our second main theorem is to show this hessian bound holds automatically. Precisely, given a connection $A$ as above we have the apriori estimate $\int_M | abla^2 F_A| < C(Λ,\dim G,M)$ for the curvature. It is important to note this result is proved in tandem with the energy quantization, and not before it. Indeed, we will in fact prove an effective version of the energy identity, and it is this effective version which will lead to both the $L^1$ hessian bound and the classical energy quantization results. In the course of the proof we will provide a quantitative version of the bubble tree decomposition which hold in all dimensions with effective estimates for a fixed stationary connections. To produce to strongest estimates in the paper we introduce an $ε$-gauge condition, which generalizes the usual Coulomb gauge and which will exist, with effective control, even over singular regions. On these $ε$-gauges we will provide a new superconvexity estimate which will be a key tool in analyzing higher dimensional annular regions.

Motivation & Objective

• To resolve the structure of the defect measure $\nu = e(x)\,d\lambda^{n-4}$ in weak limits of Yang-Mills connections.
• To establish energy quantization: $e(x)$ equals the sum of bubble energies at $x$ without additional curvature bounds.
• To prove an $L^1$-bound on $\nabla^2 F_A$ for stationary Yang-Mills connections, uniformly in curvature energy.
• To develop a quantitative bubble tree decomposition with effective estimates for fixed stationary connections.
• To introduce and utilize an $\epsilon$-gauge condition to control singular regions and enable superconvexity estimates.

Proposed method

• Introduce an $\epsilon$-gauge condition that generalizes the Coulomb gauge and allows effective control over singular regions.
• Develop a superconvexity estimate for the $L^1$ gradient of curvature on annular regions, crucial for higher-dimensional analysis.
• Construct a quantitative bubble tree decomposition via iterative covering of balls using $\delta$-weakly flat, $\delta$-bubble, and $\delta$-annular regions.
• Use the Ahlfors regularity of annular regions and transformation estimates to control curvature growth across scales.
• Prove an effective energy identity by combining the $\epsilon$-gauge estimates with the bubble tree decomposition.
• Derive the $L^1$ Hessian bound $\int_M |\nabla^2 F_A| \leq C(\Lambda, \dim G, M)$ as a consequence of the effective energy identity, not as a priori assumption.

Experimental results

Research questions

• RQ1 Can the energy density $e(x)$ of the defect measure $\nu$ be explicitly computed as the sum of energies of bubbles formed at $x$?
• RQ2 Does the $L^1$-bound on $\nabla^2 F_A$ hold automatically for stationary Yang-Mills connections with bounded energy?
• RQ3 Can a quantitative bubble tree decomposition be constructed with uniform estimates in all dimensions?
• RQ4 Can an $\epsilon$-gauge condition be defined and controlled over singular sets to enable higher-order curvature estimates?
• RQ5 Is the classical energy quantization result for Yang-Mills connections valid without assuming uniform $L^1$ Hessian bounds?

Key findings

• The energy density $e(x)$ at $n-4$ a.e. point $x$ is equal to the sum of the energies of all bubbles formed at $x$ via blow-up of a sequence of Yang-Mills connections.
• The $L^1$-bound on $\nabla^2 F_A$ holds automatically for stationary Yang-Mills connections with $\int_M |F_A|^2 \leq \Lambda$, with $\int_M |\nabla^2 F_A| \leq C(\Lambda, \dim G, M)$.
• A quantitative bubble tree decomposition is constructed with effective estimates: $\sum_c r_{c,i}^{n-4} \leq \epsilon^i$ and $\sum_a r_{a,i}^{n-4} + \sum_b r_{b,i}^{n-4} + \sum_d r_{d,i}^{n-4} \leq C(n,k,\Lambda,\delta)\sum_{j=0}^i \epsilon^j$.
• The $\epsilon$-gauge condition exists with effective control over singular regions and enables a new superconvexity estimate for the $L^1$ gradient of curvature on annular regions.
• The energy identity is proven in an effective form: the defect measure $\nu$ is quantized via bubble energies, and this implies the $L^1$ Hessian bound.
• The $L^1$ Hessian bound is derived *in tandem* with the energy identity, not as a prerequisite, via the effective bubble tree and $\epsilon$-gauge framework.