TITLE:
Energy Minimization and a Yang-Mills Gap Theorem
AUTHORS:
Raoul Bianchetti, Payam Danesh
KEYWORDS:
Yang-Mills Theory, Mass Gap, Energy Minimization, Coulomb Gauge, Flat Connections
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.7,
July
17,
2026
ABSTRACT: The theory of Yang-Mills theory possesses a notable vacuum sector consisting of flat connections, and the mass-gap conjecture asks whether the physical excitations retain a gap between the vacuum sector and the latter by an energy barrier. A naive classical reasoning just using non-vanishing curvature does not suffice, since non-flat configurations of arbitrary small energy can be attained before taking the field equations into account and gauge-fixing. In this paper, we construct an explicit energy-minimization setup which applies to gauge orbits, excludes pure-gauge dynamics, dissects out the vacuum directions in the field configuration space, and evaluates the actual orthogonal distance to the vacuum moduli space stratum. Our approach exploits Coulomb gauge fixing, Morse-Bott nature around the vacuum, elliptic coercivity of the Yang-Mills operator, critical-scale curvature bounds, and the Łojasiewicz-Simon gradient inequality. The central result shows that a smooth solution of the Yang-Mills equations on a closed Riemannian manifold with suitably small scale-critical curvature is flat. The analysis also shows the necessary Hamiltonian inequality for making the jump from the classical variational gap to a quantum mechanical spectral gap.