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

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