本书由田刚院士主编，主要介绍了几何分析领域近年来的最新研究进展，内容包括对称空间中最小曲面、具有非负Ricci曲率的完全K¨ahler流形、体积猜想、子流形、偏微分方程和黎曼几何、不变体系、几何可变体系、瞬变体系和刚片、自由度与辛几何、代数几何和物理中的超弦理论、Kaehler-Ricci孤立子唯一性，调和映射紧性，高余维平均曲率流等几何分析重点研究方向的前沿、热点。 本书适合基础数学，特别是几何分析领域的高年级本科生、研究生及教师参考。

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普林斯顿大学 北京大学几何分析1994年获美国国家科学基金委员会第19届沃特曼奖。1996年获美国数学会韦伯伦奖。2002年应邀在世界数学家大会上作一小时大会报告。

Contents
Prologue
Split-type Canonical Metrics and Related Geometric PDEs Hao Fang and Joshua Jordan 1
The Generalized Kahler Calabi-Yau Problem Xin Fu 13
Advances in the Volume Conjectures Huabin Ge and Chuwen Wang 26
Recent Progress on Austere Submanifolds Jianquan Ge and Yi Zhou 42
Multi-marginal Optimal Transport on Metric Measure Spaces Bangxian Han, Dengyu Liu, Zhuonan Zhu 56
Foliation of Area Minimizing Hypersurfaces in Asymptotically Flat Manifolds and Schoen’s Conjecture SHIHANG He, YUGUANG SHI, HAOBIN Yu 74
Classification and Uniqueness of Solutions to the Isotropic Lp Dual Minkowski Problem Haizhong Li and Yao Wan 86
Index and Total Curvature of Minimal Surfaces in Nonpositively Curved Symmetric Spaces QlONGLlNG Li 112
Recent Results on Complete Kahler Manifolds with Nonnegative Ricci Curvature Gang Liu 124
A Survey of Counterexamples to Regularity and Bernstein Property for Complex Monge-Ampere Equations ——JlAXIANG WANG AND WENLONG WANG 137
Kahler-Ricci Solitons and Optimal Degenerations of Fano Manifolds Linsheng Wang 152
The Partial Legendre Transform in Monge-Ampere Equations Ling Wang and Bin Zhou 161
On the Structure of Compact Kahler Manifolds with Nonnegative Holomorphic Sectional Curvature SHIYU ZHANG 178
Examples Related to Ricci Limit Spaces and Topology Shengxuan Zhou 186
The Existence Problem for Noncompact Completed Convex Sets in a Cone Baocheng Zhu 195
Calabi-Yau Type Theorem for Manifolds with Nonnegative Scalar Curvature Jintian Zhu 211