One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous honors, including the 1982 Fields Medal, considered the highest honor in mathematics, for his work in differential geometry. He is known also for his work in algebraic and Kähler geometry, general relativity, and string theory. His influence in the development and establishment of these areas of research has been great. These five volumes reproduce a comprehensive selection of his published mathematical papers of the years 1971 to 1991—a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kähler geometry, and general relativity. The editors have organized the contents of this collection by subject area—metric geometry and minimal submanifolds; metric geometry and harmonic functions; eigenvalues and general relativity; and Kähler geometry. Also presented are expert commentaries on the subject matter, and personal reminiscences that shed light on the development of the ideas which appear in these papers.

Comprising volumes 28 and 29 of the ALM series, this outstanding collection presents all the survey papers of Shing-Tung Yau published to date (through 2013), each with Yau's own commentary. Among these are several papers not otherwise easily accessible. Also presented are several commentaries on Yau's work written by outstanding scholars from around the world especially for publication here. Shing-Tung Yau's work is mainly in differential geometry, and he is one of the originators of the broad subject of geometric analysis - in which he remains one of the most active participants. His contributions have had an influence on both physics and mathematics, and he has long been active at the interface between geometry and theoretical physics. His proof of the positive energy theorem in general relativity demonstrated - sixty years after its discovery - that Einstein's theory is consistent and stable. His proof of the Calabi conjecture allowed physicists - using Calabi-Yau compactification - to show that string theory is a viable candidate for a unified theory of nature. Calabi-Yau manifolds are part of the standard toolkit of string theorists today. He was awarded the Fields Medal in 1982 and the Wolf Prize in 2010, and has received many other honors. The Selected Expository Works of Shing-Tung Yau with Commentary provides the reader with systematic commentary on all aspects of mathematics by a contemporary master. The reader can thereby see the world of mathematics through his particular perspective, and gain understanding of the motivation and evolution of mathematical ideas.

In this volume are collected notes of lectures delivered at the First In ternational Research Institute of the Mathematical Society of Japan. This conference, held at Tohoku University in July 1993, was devoted to geometry and global analysis. Subsequent to the conference, in answer to popular de mand from the participants, it was decided to publish the notes of the survey lectures. Written by the lecturers themselves, all experts in their respective fields, these notes are here presented in a single volume. It is hoped that they will provide a vivid account of the current research, from the introduc tory level up to and including the most recent results, and will indicate the direction to be taken by future researeh. This compilation begins with Jean-Pierre Bourguignon's notes entitled "An Introduction to Geometric Variational Problems," illustrating the gen eral framework of the field with many examples and providing the reader with a broad view of the current research. Following this, Kenji Fukaya's notes on "Geometry of Gauge Fields" are concerned with gauge theory and its applications to low-dimensional topology, without delving too deeply into technical detail. Special emphasis is placed on explaining the ideas of infi nite dimensional geometry that, in the literature, are often hidden behind rigorous formulations or technical arguments."

One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous honors, including the 1982 Fields Medal, considered the highest honor in mathematics, for his work in differential geometry. He is known also for his work in algebraic and Kähler geometry, general relativity, and string theory. His influence in the development and establishment of these areas of research has been great. These five volumes reproduce a comprehensive selection of his published mathematical papers of the years 1971 to 1991—a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kähler geometry, and general relativity. The editors have organized the contents of this collection by subject area—metric geometry and minimal submanifolds; metric geometry and harmonic functions; eigenvalues and general relativity; and Kähler geometry. Also presented are expert commentaries on the subject matter, and personal reminiscences that shed light on the development of the ideas which appear in these papers.

One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous honors, including the 1982 Fields Medal, considered the highest honor in mathematics, for his work in differential geometry. He is known also for his work in algebraic and Kähler geometry, general relativity, and string theory. His influence in the development and establishment of these areas of research has been great. These five volumes reproduce a comprehensive selection of his published mathematical papers of the years 1971 to 1991—a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kähler geometry, and general relativity. The editors have organized the contents of this collection by subject area—metric geometry and minimal submanifolds; metric geometry and harmonic functions; eigenvalues and general relativity; and Kähler geometry. Also presented are expert commentaries on the subject matter, and personal reminiscences that shed light on the development of the ideas which appear in these papers.

One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous honors, including the 1982 Fields Medal, considered the highest honor in mathematics, for his work in differential geometry. He is known also for his work in algebraic and Kähler geometry, general relativity, and string theory. His influence in the development and establishment of these areas of research has been great. These five volumes reproduce a comprehensive selection of his published mathematical papers of the years 1971 to 1991—a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kähler geometry, and general relativity. The editors have organized the contents of this collection by subject area—metric geometry and minimal submanifolds; metric geometry and harmonic functions; eigenvalues and general relativity; and Kähler geometry. Also presented are expert commentaries on the subject matter, and personal reminiscences that shed light on the development of the ideas which appear in these papers.

This volume of Surveys in Differential Geometry is dedicated to the three most eminent contributors to the subject of regularity and existence of nonlinear partial differential equations, which has played such an important role in geometry. These are Richard Hamilton, Leon Simon, and Karen Uhlenbeck. Presented topics include: analysis related to minimal submanifolds, Yang-Mills theory, Kähler metrics, Monge-Ampère equations, curve flows, and general relativity.

Geometric Analysis combines differential equations with differential geometry. An important aspect of geometric analysis is to approach geometric problems by studying differential equations. Besides some known linear differential operators such as the Laplace operator, many differential equations arising from differential geometry are nonlinear. A particularly important example is the Monge-Ampere equation. Applications to geometric problems have also motivated new methods and techniques in differential equations. The field of geometric analysis is broad and has had many striking applications. This handbook of geometric analysis presents introductions and survey papers treating important topics in geometric analysis, with their applications to related fields. It can be used as a reference by graduate students and by researchers in related areas.

In the Spring of 1984, the two authors gave a series of lectures in the Institute for Advanced Studies in Princeton. These lectures, which continued throughout the 1984-1985 academic year, are published in this volume. This greatly anticipated volume is an essential reference tool for Differential Geometry. It describes the major achievements in Differential Geometry, which progressed rapidly in the 20th century.