نسبت هندسی (مطالعات تحصیلات تکمیلی در ریاضیات)

Many problems in general relativity are essentially geometric in nature, in the sense that they can be understood in terms of Riemannian geometry and partial differential equations. This book is centered around the study of mass in general relativity using the techniques of geometric analysis. Specifically, it provides a comprehensive treatment of the positive mass theorem and closely related results, such as the Penrose inequality, drawing on a variety of tools used in this area of research, including minimal hypersurfaces, conformal geometry, inverse mean curvature flow, conformal flow, spinors and the Dirac operator, marginally outer trapped surfaces, and density theorems. This is the first time these topics have been gathered into a single place and presented with an advanced graduate student audience in mind; several dozen exercises are also included. The main prerequisite for this book is a working understanding of Riemannian geometry and basic knowledge of elliptic linear partial differential equations, with only minimal prior knowledge of physics required. The second part of the book includes a short crash course on general relativity, which provides background for the study of asymptotically flat initial data sets satisfying the dominant energy condition. Riemannian geometry: Scalar curvatureMinimal hypersurfacesThe Riemannian positive mass theoremThe Riemannian Penrose inequalitySpin geometryQuasi-local massInitial data sets: Introduction to general relativityThe spacetime positive mass theoremDensity theorems for the constraint equationsSome facts about second-order linear elliptic operatorsBibliographyIndex.