Geometric Analysis: In Honor of Gang Tian's 60th Birthday

Description

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, Sławomir Kołodziej, 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 Rivière, Yanir A. Rubinstein, Natasa Sesum, Jian Song, Jeffrey Streets, Neil S. Trudinger, Yu Yuan, Weiping Zhang, Xiaohua Zhu and Aleksey Zinger. Contents Preface A Brief Description of the Volume Big and Nef Classes, Futaki Invariant and Resolutions of Cubic Threefolds 1. Introduction 2. Futaki invariant 3. Resolutions of isolated singularities 4. Resolutions of semi-stable cubic threefolds 4.1. FΔ 4.2. FA,B References Bottom of Spectra and Amenability of Coverings 1. Introduction 2. Preliminaries 2.1. Renormalizing the Schrodinger operator 2.2. Volume comparison 2.3. Separated sets 2.4. Distance functions 2.5. Harnack inequalities 3. Modified Buser inequality 4. Back to Riemannian coverings References Some Remarks on the Geometry of a Class of Locally Conformally Flat Metrics 1. Introduction 2. Proof of Theorems 1.1 and 1.10 3. Proof of Theorem 1.12 References Analytical Properties for Degenerate Equations 0. Introduction 0.1. The arrival time 0.2. Ideas in the proof 1. Gradient flows in finite dimensions 1.1. Lojasiewicz inequalities 1.2. Arnold–Thom conjectures 2. Lojasiewicz theorem for the arrival time 2.1. The flow lines approach the critical set orthogonally 3. Theorem 0.2 and an estimate for rescaled MCF 3.1. Rescaled mean curvature flow 3.2. Rate of convergence of the rescaled MCF 3.3. A strong cylindrical approximation 3.4. Reduction 3.5. The summability condition (3.11) 4. Approximate eigenfunctions on cylinders 4.1. Eigenfunctions on cylinders 4.2. The frequency References Equivariant K-theory and Resolution I: Abelian Actions Introduction 1. Resolution 2. Lifting 3. Reduction 4. Reduced K-theory 5. Delocalized equivariant cohomology 6. The relative sequences 7. The isomorphism 8. Examples References On the Existence Problem of Einstein–Maxwell Kahler Metrics 1. Introduction 2. Volume minimization for Einstein–Maxwell Kahler metrics 3. The normalized Einstein–Hilbert functional 4. The normalized Einstein–Hilbert functional for toric Kahler manifolds 5. Toric K-stability References Local Moduli of Scalar-flat Kahler ALE Surfaces 1. Introduction 1.1. Deformations of the minimal resolution 2. Construction of the local moduli space 2.1. Outline of Proof of Theorem 2.3 2.2. Universality 3. The case of the minimal resolution 3.1. Cyclic quotient singularity 3.2. Non-cyclic quotient singularities 4. Dimension of the moduli space 4.1. Discussion of Table 1.1 4.2. Cyclic case 4.3. Non-cyclic cases 4.4. Hyperk¨ahler case 5. Appendix References Singular Ricci Flows II 1. Introduction 2. Notation and terminology 3. Compact K-solutions 4. Curvature and volume estimates 5. Asymptotic conditions 6. Dimension of the set of singular times References An Inequality Between Complex Hessian Measures of Holder Continuous m-subharmonic Functions and Capacity 1. Introduction 2. Preliminaries 3. The Dirichlet problem References A Guided Tour to Normalized Volume 1. Introduction 1.1. History 1.2. Outline 2. Definitions and first properties 2.1. Definitions 2.2. Properties 3. Stability in Sasaki–Einstein geometry 3.1. T-varieties 3.2. K-stability 3.3. Sasaki–Einstein geometry 4. Stable degeneration conjecture 4.1. Statement 4.2. Cone case 4.2.1. Rank one case 4.2.2. Log Fano cone in general 4.2.3. Uniqueness 4.3. Results on the general case 5. Applications 5.1. Equivariant K-semistability of Fano 5.2. Donaldson–Sun’s Conjecture 5.2.1. K-semistable degeneration 5.2.2. Uniqueness of polystable degeneration 5.3. Estimates in dimension three and K-stability of threefolds 6. Questions and future research 6.1. Revisit stable degeneration conjecture 6.2. Birational geometry study 6.2.1. Inversion of adjunction 6.2.2. Uniform bound 6.3. Miscellaneous questions 6.3.1. Positive characteristics 6.3.2. Relation to local orbifold Euler numbers 6.3.3. Normalized volume function References Towards a Liouville Theorem for Continuous Viscosity Solutions to Fully Nonlinear Elliptic Equations in Conformal Geometry 1. Introduction 2. The strong comparison principle and the Hopf Lemma 2.1. Proof of the strong comparison principle 2.2. Proof of the Hopf Lemma 3. Proof of the Liouville theorem References Arsove–Huber Theorem in Higher Dimensions 1. Introduction: the story in two dimensions 2. n-Laplace equations as higher-dimensional analogues 2.1. Introduction of n-Laplace equations in conformal geometry 2.2. Non-linear potential theory for n-Laplace equations 2.3. Isolated singularity for nonnegative n-superharmonic functions 2.4. Higher-dimensional analogue of Arsove–Huber estimates 2.5. Higher-dimensional analogue of Taliaferro’s estimates 3. n-Laplace equations in conformal geometry 4. Hypersurfaces in hyperbolic space References From Local Index Theory to Bergman Kernel: A Heat Kernel Approach 0. Introduction 1. Local index theorem 1.1. Chern–Weil Theory 1.2. Atiyah–Singer index theorem 1.3. Heat kernel and McKean–Singer formula 1.4. Proof of the local index theorem 2. Holomorphic Morse inequalities 3. Bergman kernels 3.1. Asymptotic expansion of Bergman kernels 3.2. Proof of the asymptotic expansion of Bergman kernels 3.3. Coefficients of the asymptotic expansion of Bergman kernels References Fourier–Mukai Transforms, Euler–Green Currents, and K-Stability 1. Introduction and statement of results 1.1. Hermitean metrics and base change 2. Classical elimination theory 3. Linear algebra of complexes and the torsion of a exact complex 4. Fourier–Mukai transforms and the geometric technique 4.1. The basic set up for resultants 4.2. The basic set up for discriminants 5. Comparing the currents δZ and δI over S References The Variations of Yang–Mills Lagrangian I. Introduction II. The Plateau problem II.1. The conformal parametrization choice as a Coulomb gauge III. A Plateau type problem on the lack of integrability III.1. Horizontal equivariant plane distributions III.1.1. The definition III.1.2. Characterizations of equivariant horizontal distribution of plane by1-forms on Bm taking values into G. III.2. The lack of integrability of equivariant horizontal distribution of planes III.3. The gauge invariance III.4. The Coulomb gauges IV. Uhlenbeck’s Coulomb gauge extraction method IV.1. Uhlenbeck’s construction IV.2. A refinement of Uhlenbeck’s Coulomb gauge extraction theorem IV.3. Controlled gauges without small energy assumption V. The resolution of the Yang–Mills Plateau problem in the critical dimension V.1. The small energy case V.2. The general case and the point removability result for W1,2 Sobolev connections VI. The Yang–Mills equation in sub-critical and critical dimensions VI.1. Yang–Mills fields VI.2. The regularity of W1,2 Yang–Mills fields in sub-critical and critical dimensions VII. Concentration compactness and energy quantization for Yang–Mills fields in critical dimension VIII. The resolution of the Yang–Mills Plateau problem in super-critical dimensions VIII.1. The absence of W1,2 local gauges VIII.2. Tian’s results on the compactification of the space of smooth Yang–Mills fields in high dimensions VIII.3. The Ω-anti-self-dual instantons VIII.4. Tian’s regularity conjecture on Ω-anti-self-dual instantons VIII.5. The space of weak connections VIII.6. The resolution of the Yang–Mills Plateau problem in five dimensions VIII.7. Weak holomorphic structures over complex manifolds References Tian’s Properness Conjectures: An Introduction to Kahler Geometry Prologue A second prologue 1. Introduction 2. Kahler and Fano manifolds 3. The Mabuchi energy 3.1. The K-energy when μ < 0 3.2. The K-energy when μ ≥ 0 3.3. Tian’s invariant 4. The Kahler–Einstein equation 5. Properness implies existence 5.1. A two-parameter continuity method 5.2. Openness 5.3. An L∞ bound in the sub-rectangle 5.4. An L∞ bound in the interval 5.5. Second-order estimates 5.6. Higher-order compactness via Evans–Krylov’s estimate 5.7. Properness implies existence 6. A counterexample to Tian’s first conjecture and a revised conjecture 6.1. Why Tian’s conjecture is plausible 6.2. A counterexample 7. Infinite-dimensional metrics on H 8. Metric completions of H 8.1. The Calabi metric completion 8.2. The Mabuchi metric completion 9. The Darvas metric and its completion 10. The Aubin functional and the Darvas distance function 11. Quotienting the metric completion by a group action 11.1. The action of the automorphism group on H 11.2. The Aubin functional on the quotient space 12. A modified conjecture 13. A general existence/properness principle 14. Applying the general existence/properness principle 15. A proof of Tian’s modified first conjecture 16. A proof of Tian’s second conjecture: the Moser–Trudinger inequality References Ancient Solutions in Geometric Flows 1. Introduction 2. Ancient solutions to the Ricci flow 2.1. Ancient closed solutions 2.2. Complete ancient solutions 3. Ancient solutions to the Mean Curvature Flow 3.1. Curve shortening flow 3.2. Closed ancient solutions to the MCF 3.3. Complete ancient solutions to the MCF 3.4. Sketch of the proof of Theorem 3.11 4. Ancient solutions to Yamabe flow References The Kahler–Ricci Flow on CP2 1. Introduction 2. Proof References Pluriclosed Flow and the Geometrization of Complex Surfaces 1. Introduction Part I: Pluriclosed flow 2. Existence and basic regularity properties 2.1. Definition and local existence 2.2. Pluriclosed flow as a gradient flow 3. Conjectural existence properties 3.1. Sharp local existence for Kahler–Ricci flow 3.2. A positive cone and conjectural existence for pluriclosed flow 3.3. Characterizations of positive cones 4. (1,0)-form reduction 5. Pluriclosed flow of locally homogeneous surfaces 5.1. Wall’s classification 5.2. Existence and convergence results Part II: Geometrization of complex surfaces 6. Conjectural limiting behavior on K¨ahler surfaces 6.1. Surfaces of general type 6.2. Properly Elliptic surfaces 6.3. Elliptic surfaces of Kodaira dimension zero 6.4. Rational and ruled surfaces 7. Conjectural limiting behavior on non-Kahler surfaces 7.1. Properly elliptic surfaces 7.2. Kodaira surfaces 7.3. Class VII0 surfaces 7.3.1. Inoue surfaces. 7.3.2. Hopf surfaces. 7.4. Class VII+ surfaces Part III: Classification of generalized Kahler structures 8. Generalized Kahler geometry 9. Generalized Kahler–Ricci flow 9.1. Commuting case 9.2. Nondegenerate case 9.3. General case References From Optimal Transportation to Conformal Geometry 1. Introduction 2. Optimal transportation 3. Augmented Hessian equations 4. Application to conformal geometry References Special Lagrangian Equations 1. Introduction 1.1. Definition of the equation 1.2. Special Lagrangian submanifold background of the equation 1.3. Algebraic form of the equation 1.4. Level set of the equation 2. Results 2.1. Outline 2.2. Rigidity of entire solutions 2.3. A priori estimates for Monge–Ampere equation 2.4. A priori estimates for special Lagrangian equation with critical and supercritical phases 2.5. Singular solutions to special Lagrangian equation with subcritical phase 3. Curvature flows with potential 3.1. Lagrangian mean curvature flow in Euclidean space 3.2. Lagrangian mean curvature flow in pseudo-Euclidean space and Kahler–Ricci flow on Kahler manifold 4. Problems References Positive Scalar Curvature on Foliations: The Enlargeability 0. Introduction 1. Proof of Theorem 0.2 1.1. The case of compactly enlargeable foliations 1.2. The case where M is noncompact References Kahler–Einstein Metrics on Toric Manifolds and G-manifolds 0. Introduction 1. Preliminary on toric manifolds 2. A priori C0-estimate 3. Generalization of Lemma 2.1 and its applications 3.1. Deformation of Ricci flow 3.2. Singular solutions arising in the continuity method 4. Reduced K-energy μ(u) 4.1. The reduction of K-energy 4.2. Properness of K(φ) 5. Kahler–Einstein metrics on G-manifolds 5.1. Reduced scalar curvature equation on a+ 5.2. A sketch of proof of Theorem 5.2 5.3. Proof of the necessary part of Theorem 5.1 6. Appendix: Examples of Fano G-manifolds References Some Questions in the Theory of Pseudoholomorphic Curves 1. Topology of moduli spaces 2. BPS states for arbitrary symplectic manifolds 3. Symplectic degenerations and Gromov–Witten invariants 4. Geometric applications References