A particularly important example is the Monge-Ampre 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.

Moduli Spaces of Projective ManifoldsGeometric Analysis combines differential equations and differential geometry. An important aspect is to solve 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-Ampre 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 provides introductions to and surveys of important topics in geometric analysis and their applications to related fields which is intend to be referred by graduate students and researchers in related areas.

Numerical Approximations to Extremal Metrics on Toric Surfaces

 R. S. Bunch, Simon K. Donaldson I

 1 Introduction

 2 The set-up

 2.1 Algebraic metrics

 2.2 Decomposition of the curvature tensor

 2.3 Integration

 3 Numerical algorithms: balanced metrics and refined approximations.

 4 Numerical results

 4.1 The hexagon

 4.2 The pentagon

 4.3 The octagon

 4.4 The heptagon

 5 Conclusions

 References

Kiihler Geometry on Toric Manifolds, and some other Manifolds with Large Symmetry

Glning Constructions of Special Lagrangian Cones

Harmonic Mappings

Harmonic Functions on Complete Riemannian Manifolds

Complexity of Solutions of Partial Differential Equations

Variational Principles on Triangulated Surfaces

Asymptotic Structures in the Geometry of Stability and Extremal Metrics

Stable Constant Mean Curvature Surfaces

A General Asymptotic Decay Lemma for Elliptic Problems

Uniformization of Open Nonnegatively Curved K／ihler Manifolds in Higher Dimensions

Geometry of Measures：Harmonic Analysis Meets Geometric Measure Theory

The Monge Ampere Eequation and its Geometric Aapplications

Lectures on Mean Curvature Flows in Higher Codimensions

Local and Global Analysis of Eigenfunctions on Riemannian Manifolds

Yau’S Form of Schwarz Lemma and Arakelov Inequality On