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Softcover ISBN:  9781470466237 
Product Code:  GSM/201.S 
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Book DetailsGraduate Studies in MathematicsVolume: 201; 2019; 361 ppMSC: Primary 53; 83
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.
ReadershipGraduate students and researchers interested in nonlinear differential equations and, in particular, in mathematical aspects of general relativity.

Table of Contents

Riemannian geometry

Scalar curvature

Minimal hypersurfaces

The Riemannian positive mass theorem

The Riemannian Penrose inequality

Spin geometry

Quasilocal mass

Initial data sets

Introduction to general relativity

The spacetime positive mass theorem

Density theorems for the constraint equations

Some facts about secondorder linear elliptic operators


Additional Material

Reviews

'Geometric Relatively' is refreshing in its narrative approach to this topic. The author is open and honest about the material included and the material excluded in the text, explaining when certain material is omitted or glossed over. Indeed, oftentimes finer technical details will be omitted from a proof for the sake of narrative clarity. Overall, this book is a nice textbook for a graduate student to study from or a great reference for a research mathematician. Anyone who is interested in exploring relativity from a geometry perspective or simply interested purely in geometric analysis can gain something from this text.
John Ross, Southwestern University


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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.
Graduate students and researchers interested in nonlinear differential equations and, in particular, in mathematical aspects of general relativity.

Riemannian geometry

Scalar curvature

Minimal hypersurfaces

The Riemannian positive mass theorem

The Riemannian Penrose inequality

Spin geometry

Quasilocal mass

Initial data sets

Introduction to general relativity

The spacetime positive mass theorem

Density theorems for the constraint equations

Some facts about secondorder linear elliptic operators

'Geometric Relatively' is refreshing in its narrative approach to this topic. The author is open and honest about the material included and the material excluded in the text, explaining when certain material is omitted or glossed over. Indeed, oftentimes finer technical details will be omitted from a proof for the sake of narrative clarity. Overall, this book is a nice textbook for a graduate student to study from or a great reference for a research mathematician. Anyone who is interested in exploring relativity from a geometry perspective or simply interested purely in geometric analysis can gain something from this text.
John Ross, Southwestern University