Lectures on Mathematical Relativity

This book introduces and explores some of the deep connections between Einstein's theory of gravitation and differential geometry. As an outgrowth of graduate summer schools, the presentation is aimed at graduate students in mathematics and mathematical physics, starting from the foundations of special and general relativity, and moving to more advanced results in geometric analysis and the Einstein constraint equations. Topics include the formulation of the Einstein field equation and the Einstein constraint equations; gluing construction of initial data sets which are Schwarzschild near infinity; and an introduction to the Riemannian Penrose inequality. While the book assumes a background in differential geometry and real analysis, a number of basic results in geometry are provided. There are well over 100 exercises, many woven into the fabric of the chapters as well as others collected at the end of chapters, to give readers a chance to engage and extend the text.

Justin Corvino is Marshall R. Metzgar Professor in the Department of Mathematics at Lafayette College, Pennsylvania. He earned his Ph.D. at Stanford in 2000 under the direction of Richard M. Schoen after completing his undergraduate degree at the Massachusetts Institute of Technology (MIT). He was previously an NSF Postdoctoral Research Fellow and Tamarkin Assistant Professor at Brown University. He spent the fall of 2008 at the Mittag-Leffler Institute in Sweden as a Fulbright Fellow. He has authored or co-authored numerous journal articles in geometric analysis and mathematical physics. Pengzi Miao is a professor in the Department of Mathematics at the University of Miami. He was previously a lecturer at Monash University in Melbourne, Australia. He obtained his Ph.D. from Stanford in 2003 with advisor Richard M. Schoen, after which he worked as a postdoctoral fellow at the Mathematical Sciences Research Institute in Berkeley, California in 2004, and as a Visiting Assistant Professor at UC Santa Barbara from 2004 to 2006. He has authored or co-authored dozens of research articles in geometric analysis and general relativity.