This edited volume has a two-fold purpose. First, comprehensive survey articles provide a way for beginners to ease into the corresponding sub-fields. These are then supplemented by original works that give the more advanced readers a glimpse of the current research in geometric analysis and related PDEs.
The book is of significant interest for researchers, including advanced Ph.D. students, working in geometric analysis. Readers who have a secondary interest in geometric analysis will benefit from the survey articles.
The results included in this book will stimulate further advances in the subjects: geometric analysis, including complex differential geometry, symplectic geometry, PDEs with a geometric origin, and geometry related to topology.
Contributions by Claudio Arezzo, Alberto Della Vedova, Werner Ballmann, Henrik Matthiesen, Panagiotis Polymerakis, Sun-Yung A. Chang, Zheng-Chao Han, Paul Yang, Tobias Holck Colding, William P. Minicozzi II, Panagiotis Dimakis, Richard Melrose, Akito Futaki, Hajime Ono, Jiyuan Han, Jeff A. Viaclovsky, Bruce Kleiner, John Lott, Slawomir Kolodziej, Ngoc Cuong Nguyen, Chi Li, Yuchen Liu, Chenyang Xu, YanYan Li, Luc Nguyen, Bo Wang, Shiguang Ma, Jie Qing, Xiaonan Ma, Sean Timothy Paul, Kyriakos Sergiou, Tristan Riviere, Yanir A. Rubinstein, Natasa Sesum, Jian Song, Jeffrey Streets, Neil S. Trudinger, Yu Yuan, Weiping Zhang, Xiaohua Zhu and Aleksey Zinger.
Preface.- Big and nef classes, Futaki Invariant and resolutions of cubic threefolds.- Bottom of spectra and amenability of coverings.- Some remarks on the geometry of a class of locally conformally at metrics.- Analytical properties for degenerate equations.- On the existence problem of Einstein-Maxwell Kahler Metrics.- Local moduli of scalar-flat Kahler ale surfaces.- Singular Ricci flows II.- An inequality between complex Hessian measures of Hoelder continuous m-subharmonic functions and capacity.- A guided tour to normalized volume.- Towards a Liouville theorem for continuous viscosity solutions to fully nonlinear elliptic equations in conformal geometry.- Equivariant K-theory and Resolution I: Abelian actions.- Arsove-Huber's Theorem in Higher Dimensions.- From local index theory to Bergman kernel: a heat kernel approach.- Fourier-Mukai Transforms, Euler-Green Currents, and K-Stability.- The Variations of Yang-Mills Lagrangian.- Tian's properness conjectures: an introduction to Kahler geometry.- Ancient solutions in geometric flows.- The Kahler-Ricci flow on CP2.- Pluriclosed flow and the geometrization of complex surfaces.- From Optimal Transportation to Conformal Geometry.- Special Lagrangian Equation.- Positive scalar curvature on foliations: the enlargeability.- Kahler-Einstein metrics on toric manifolds and G-manifolds.- Some Questions in the Theory of Pseudoholomorphic Curves.