Geometrically defined surfaces in general relativity and their relation to physical invariants

This thesis explores the intersection of differential geometry, geometric analysis, and general relativity, focusing on geometric structures and their physical implications. The primary objective is to deepen the understanding of quasi-local energy definitions, particularly the Hawking energy, and to develop geometric methods that illuminate fundamental problems in mathematical relativity. In the first part, we analyze local foliations of prescribed mean curvature surfaces, such as spacetime constant mean curvature (STCMC) and constant expansion (CE) surfaces. These surfaces are instrumental in characterizing the center of mass in asymptotically flat spacetimes. Using a Lyapunov-Schmidt reduction framework, we construct local foliations around a point, prove their uniqueness, and establish conditions for their nonexistence. The second part focuses on one of the central challenges in the field: identifying a viable quasi-local energy. We investigate the Hawking energy and demonstrate that when evaluated on specific critical surfaces, referred to as "Hawking surfaces", it satisfies key physical properties, including positivity, correct asymptotics, monotonicity, and rigidity, provided certain technical conditions are met. These results confirm the Hawking energy's consistency with fundamental physical principles and address long-standing ambiguities and criticisms. Additionally, we study foliations by Hawking surfaces in both local and large-scale settings. In both cases, we employ a Lyapunov-Schmidt reduction to establish the existence and uniqueness of the foliations. For large-scale foliations, we analyze the center of the foliations and their physical significance. In the local setting, we compare the foliation's behavior to the small sphere limit along a null cone, resolving apparent discrepancies and refining the physical interpretation. Through these results, we aim to highlight the Hawking energy's practical and theoretical importance as a quasi-local energy , redeeming its reputation and reinforcing its relevance in gravitational physics.