Stability for the Positive Mass Theorem and the Penrose Inequality

This dissertation investigates the stability of two fundamental inequalities in general relativity and differential geometry: the Positive Mass Theorem and the Riemannian Penrose Inequality. The Positive Mass Theorem states that an asymptotically flat 3-manifold with non-negative scalar curvature must have non-negative mass, and the mass is zero only for Euclidean space. The Penrose Inequality, concerning asymptotically flat 3-manifolds with an outermost minimal boundary, posits a relationship between the mass and the area of the boundary, with equality holding for the Schwarzschild manifold. This work focuses on the stability aspects of these theorems, examining how manifolds behave when their mass is close to the critical values defined by these inequalities. Specifically, it demonstrates that under certain conditions, such manifolds are topologically close to Euclidean space or the Schwarzschild manifold, modulo minor perturbations. The research explores the geometric properties of these manifolds, employing concepts like the pointed measured Gromov-Hausdorff topology. It analyzes scenarios where the mass is almost zero, showing that the manifold approximates Euclidean 3-space. Additionally, it investigates the case where the mass is close to the lower bound given by the Penrose inequality, demonstrating proximity to the Schwarzschild manifold. This stability analysis is crucial for understanding the robustness of these geometric and physical principles.

Curriculum

This dissertation is structured into sections covering preliminary concepts, stability for the Positive Mass Theorem, and stability for the Penrose Inequality. The preliminary section lays the groundwork with basic Riemannian geometry, the Schwarzschild metric, and asymptotic flatness.

1. Preliminary
   • Basics of Riemannian geometry2.0.2
   • Geometry of Schwarzschild metric2.0.3
   • Asymptotically flat 3-manifolds2.0.4
   • pm-GH convergence modulo negligible spikes2.0.5
2. Stability for the Positive Mass Theorem
   • Regular region with small area boundary3.1
   • GH convergence modulo negligible spikes in general dimensions3.3
3. Stability for the Penrose Inequality
   • Capacity of horizon and Green’s function4.1
   • Integral estimate and weighted volume comparison4.2
   • Harmonic coordinate for conformal metric4.3
   • W¹,²-convergence of elliptic equations4.4
   • Proof of Theorems 1.0.4 and 1.0.54.5

Careers

A PhD in Mathematics from Stony Brook University equips graduates with advanced analytical, problem-solving, and research skills, preparing them for diverse career paths in academia and industry.

• University Professor/Researcher
• Data Scientist
• Quantitative Analyst
• Cryptographer
• Mathematician in Research & Development