Handbook of Geometric Analysis, Vol. 1 (Advanced Lectures in Mathematics No. 7)

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-Amper? 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 the first of the two to be published in the ALM series 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. Table of contents Numerical Approximations to Extremal Metrics on Toric Surfaces (R. S. Bunch, Simon K. Donaldson) K?hler Geometry on Toric Manifolds, and some other Manifolds with Large Symmetry (Simon K. Donaldson) Gluing Constructions of Special Lagrangian Cones (Mark Haskins, Nikolaos Kapouleas) Harmonic Mappings (J?rgen Jost) Harmonic Functions on Complete Riemannian Manifolds (Peter Li) Complexity of Solutions of Partial Differential Equations (Fang Hua Lin) Variational Principles on Triangulated Surfaces (Feng Luo) Asymptotic Structures in the Geometry of Stability and Extremal Metrics (Toshiki Mabuchi) Stable Constant Mean Curvature Surfaces (William H. Meeks III, Joaqu?n P?rez, Antonio Ros) A General Asymptotic Decay Lemma for Elliptic Problems (Leon Simon) Uniformization of Open Nonnegatively Curved K?hler Manifolds in Higher Dimensions (Luen-Fai Tam) Geometry of Measures: Harmonic Analysis Meets Geometric Measure Theory (Tatiana Toro) Lectures on Mean Curvature Flows in Higher Codimensions (Mu-Tao Wang) Local and Global Analysis of Eigenfunctions on Riemannian Manifolds (Steve Zelditch) Yau's Form of Schwarz Lemma and Arakelov Inequality On Moduli Spaces of Projective Manifolds (Kang Zuo)