This volume is an account on recent advances in geometric analysis and related equations, including Ricci flow, affine normal flow, geometric analysis on pseudoconvex hypersurfaces, Alexandrov space, manifolds with special holonomy, and singular plateau problem. Many of the articles are distributed by leading experts in the field, and were presented at the 2007 International Conference in Geometric Analysis. This volume is intended for researchers and graduate students in geometric analysis and related areas.

The launch of this Advanced Lectures in Mathematics series is aimed at keeping mathematicians informed of the latest developments in mathematics, as well as to aid in the learning of new mathematical topics by students all over the world. Each volume consists of either an expository monograph or a collection of significant introductions to important topics. This series emphasizes the history and sources of motivation for the topics under discussion, and also gives an overview of the current status of research in each particular field. These volumes are the first source to which people will turn in order to learn new subjects and to discover the latest results of many cutting-edge fields in mathematics.

Preface

Huai-Dong Cao: Recent Progress on Ricci Solitons

Jianguo Cao, Bo Dai, and Jiaqiang Mei: An Optimal Extension of Perelman's Comparison Theorem for Quadrangles and its Applications

Der-Chen Chang and Stephen S.-T. Yau: Geometric Analysis on a Family of Pseudoconvex Hypersurfaces

Shanyu Ji: A New Proof for Faran's Theorem on Maps Between B2 and B3

Spiro Karigiannis: Some Notes on G2 and Spin(7) Geometry

Lei Ni: Closed Type I Ancient Solutions to Ricci Flow

John Loftin and Mao-Pei Tsui: Limits of Solutions to a Parabolic Monge-Ampere Equation

Chang-Shou Lin and Chin-Lung Wang: A Function Theoretic View of the Mean Field Equations on Tori

Siu-Cheong Lau and Naichung Conan Leung: Conformal Geometry and Special Holonomy

Luen-Fai Tam: Exhaustion Functions on Complete Manifolds

Sumio Yamada: On Singular Plateau Problem