Yang-Mills theory on a fractal manifold with boundary (PART C) Stability analysis of solutions to the fractal Yang-Mills equations

This paper provides a detailed analysis of the stability of solutions to the Yang?Mills equations defined on fractal manifolds. Extending the traditional Yang-Millstheory to fractal manifolds with Hausdorff dimension DH, we investigate the sta?bility behavior of solutions under small perturbations. The Yang-Mills equationson fractal manifolds are derived as the variational equations of the energy func?tional. Through second-order variational analysis, we verify whether the solutionscorrespond to stable minima. By leveraging the fractal Sobolev inequalities andproperties of the fractal Laplace operator, we demonstrate that the solutions ex?hibit both mathematical and dynamical stability. These findings lay a theoreticalfoundation for research into quantum field theories in fractal geometries