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This is a Quantitative Study study.

October 12, 2025Open Access

The quantitative behavior of α-Yang-Mills-Higgs fields on surfaces

Key Points

The research establishes the α-energy identity and demonstrates the no-neck property in the blow-up regime.
Key findings include the identification of a hidden Jacobian structure through Hodge decomposition, enhancing understanding of energy density control.
The analysis improves Lorentz space estimates, refining from L2,∞ to the optimal L2,1 scale.
These results unify concepts in singularity formation within variational gauge theories, indicating broad implications for mathematical physics.

Abstract

We investigate the blow-up behavior of α-Yang--Mills--Higgs (α-YMH) fields over closed Riemannian surfaces with the target fiber F = S^K-1 RK being the round sphere, focusing on the establishment of the α-energy identity and the no-neck property during the bubbling process. A central innovation is the identification of a hidden Jacobian structure through Hodge decomposition and a new conservation law. Furthermore, we derive a Pohozaev-type identity for α-YMH fields, which enables refined control of the energy density. Together, these advances ensure the validity of the α-energy identity as α 1. Our analysis further sharpens the Lorentz space estimates from the L^2, to the optimal L^2, 1 scale, ultimately yielding the no-neck property in the blow-up regime. These results provide a unified and quantitative framework for understanding singularity formation in variational gauge theories.

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