Abstract
The Positive Mass Theorem states that a complete asymptotically flat manifold with nonnegative scalar curvature has nonnegative total mass, and the Riemannian Penrose Inequality relates the total mass to the area of the outermost minimal surface. Recent developments, including the mu-bubble method, have provided new tools for studying scalar curvature and geometric inequalities.
This dissertation investigates geometric inequalities and horizon existence under scalar curvature constraints using the mu-bubble construction, together with capacity theory and quasi-local mass comparisons.
In the first part, we study asymptotically flat three-manifolds with two ends modeled on three-dimensional Euclidean space with the origin removed. We introduce an outward minimizing capacity invariant and establish inequalities relating it to the total mass using mu-bubble constructions. These results yield sufficient conditions for the existence of horizons. We further analyze the rigidity case and obtain a strengthened rigidity result extending the classical equality case of the Penrose inequality, along with criteria characterizing two-sided Schwarzschild manifolds.
In the second part, which forms the main focus of this dissertation, we study topological cylindrical manifolds with boundary of the form a closed surface times a half-line. Using mu-bubble constructions, we compare quasi-local mass functionals, including the Bartnik and Brown–York masses, with an asymptotic area threshold. We prove that suitable geometric conditions on the boundary imply the existence of an interior minimal surface. This provides a new mechanism for detecting horizons that is distinct from classical variational and mean curvature flow methods.