Critical metrics on 4 manifolds with harmonic anti self dual weyl tensor

Understanding the geometric structure of Riemannian manifolds plays a central role not only in pure mathematics but also in theoretical and computational physics, where curvature-driven models underpin fields ranging from general relativity to quantum field theory. In this work, we investigate 4-dimensional simply connected compact Miao–Tam critical metrics of the volume functional under the additional assumption of a harmonic Anti-Self Dual (ASD) Weyl tensor. Such metrics arise naturally as stationary points of the volume functional restricted to metrics of fixed scalar curvature with prescribed boundary geometry, forming a geometric variational problem whose analytical structure echoes algorithmic optimization frameworks widely used in computational modeling.Four-dimensional manifolds occupy a distinguished position in both mathematics and physics due to the decomposition of the bundle of 2-forms into self-dual and anti-self dual components. This feature allows one geometric control of curvature and parallels techniques in theoretical physics where duality and tensor decompositions play key roles in gauge theories and gravitational models. Building upon this structure, we establish rigidity results for Miao–Tam critical metrics satisfying a sharp pinching condition involving the Ricci curvature and its projection along the gradient of the potential function. Through rewned curvature identities and divergence formulas, we show that any such manifold must be isometric to a geodesic ball in one of the simply connected space forms of dimensional 4. Beyond the intrinsic geometric interest, these results contribute to broader computational–theoretical physics by clarifying the space of admissible geometric configurations for curvature-constrained variational models. Since curvature functionals are central to numerical. Relativity, geometric rows, and high-dimensional optimization schemes, the classiffcation of critical points under harmonicity conditions offers guidance for computational algorithms seeking stable solutions across scales from local geometric data to global manifold structure. The harmonic ASD Weyl condition, in particular, connects to energy minimization principles frequently used in discretized geometric simulations. Thus, this work not only advances the understanding of critical metrics in differential geometry but also aligns with cross-disciplinary modeling efforts that leverage analytical frameworks to bridge mathematical structures with computational and physical applications.