Explore a 55-minute lecture on the stability of critical points of the self dual abelian YMH energy on Kähler manifolds, delivered by Daniel Fadel from UFRJ. Delve into the intricate world of manifolds with special geometric structures, examining their relationships with complex and algebraic geometry, global analysis, theoretical physics, and symplectic geometry. Investigate topics such as Monge-Ampère type equations, special holonomy, quaternionic geometry, twistor theory, non-Kähler complex manifolds, harmonic maps, Einstein and soliton metrics, homogeneous spaces, integrable systems, gauge theory, geometric flows, and mathematical string- and M-theory. Gain insights from this presentation, part of a broader conference organized by esteemed mathematicians from various institutions, held at IMPA in Rio de Janeiro.