[Paper Review] The energy identity for a sequence of Yang-Mills α-connections

Min-Chun Hong, Lorenz Schabrun|arXiv (Cornell University)|Aug 12, 2013

Geometric Analysis and Curvature Flows18 references3 citations

TL;DR

This paper establishes the energy identity for a sequence of Yang-Mills α-connections as α→1, proving that the total energy splits into the limit connection's energy and the energies of bubble maps from S². Using the Palais-Smale condition for the α-functional, the authors show convergence away from finitely many points and derive an energy identity analogous to the Sacks-Uhlenbeck program in harmonic maps, extending it to Yang-Mills theory in four dimensions.

ABSTRACT

We prove that the Yang-Mills $α$-functional satisfies the Palais-Smale condition. This guarantees the existence of critical points, which are called Yang-Mills $α$-connections. It was shown by Hong, Tian and Yin in [10] (to appear in Comm. Math. Helv.) that as $α o 1$, a sequence of Yang-Mills $α$-connections converges to a Yang-Mills connection away from finitely many points. We prove an energy identity for such a sequence of Yang-Mills $α$-connections. As an application, we also prove an energy identity for the Yang-Mills flow at the maximal existence time.

Motivation & Objective

• To establish the energy identity for sequences of Yang-Mills α-connections as α→1, extending the Sacks-Uhlenbeck approach from harmonic maps to Yang-Mills theory.
• To prove that the Yang-Mills α-functional satisfies the Palais-Smale condition, ensuring existence of critical points (Yang-Mills α-connections).
• To analyze the blow-up behavior of such sequences and show that energy is conserved in the limit, accounting for bubble formation.
• To apply the energy identity to the Yang-Mills flow, proving energy identity at the maximal existence time.

Proposed method

• Prove the Palais-Smale condition for the Yang-Mills α-functional $ YM_\alpha(A) = \int_M (1 + |F_A|^2)^\alpha \, dV $, ensuring existence of critical points.
• Use the Yang-Mills α-connection equation $ D_A^*\left((1 + |F_A|^2)^{\alpha-1} F_A\right) = 0 $ as the Euler-Lagrange equation.
• Apply Uhlenbeck's weak compactness and gauge-fixing techniques to control curvature and connection forms in local coordinates.
• Establish decay estimates via Morrey-type inequalities and Sobolev embeddings to prove Hölder continuity and smoothness of connections.
• Use blow-up analysis and rescaling to identify bubble maps $ \omega_i: S^2 \to N $ at concentration points.
• Derive the energy identity $ \lim_{\alpha \to 1} YM_\alpha(A_\alpha) = YM(A_\infty) + \sum_{i=1}^l E(\omega_i) $ via compactness and energy quantization.

Experimental results

Research questions

• RQ1 Does the energy identity hold for a sequence of Yang-Mills α-connections as α→1, analogous to the Sacks-Uhlenbeck result for harmonic maps?
• RQ2 Can the Palais-Smale condition be established for the Yang-Mills α-functional in four dimensions?
• RQ3 What is the structure of the blow-up profile for Yang-Mills α-connections near concentration points?
• RQ4 Does the energy identity extend to the Yang-Mills flow at its maximal existence time?
• RQ5 How do the energies of bubble maps from S² contribute to the total energy in the limit?

Key findings

• The Yang-Mills α-functional satisfies the Palais-Smale condition for all α>1, guaranteeing the existence of critical points (Yang-Mills α-connections).
• A sequence of Yang-Mills α-connections converges smoothly to a Yang-Mills connection away from finitely many points as α→1.
• The energy identity holds: $ \lim_{\alpha \to 1} YM_\alpha(A_\alpha) = YM(A_\infty) + \sum_{i=1}^l E(\omega_i) $, where $ \omega_i $ are bubble maps from $ S^2 $.
• The blow-up analysis yields that the total energy splits into the energy of the limit connection and the energies of finitely many harmonic spheres.
• The energy identity is extended to the Yang-Mills flow, proving conservation of energy at the maximal existence time.
• The proof relies on sharp decay estimates, gauge-fixing, and Morrey-type inequalities to establish Hölder continuity and smoothness of the limiting connection.