前辅文
Numerical Approximations to Extremal Metrics on Toric Surfaces
R. S. Bunch, Simon K. Donaldson.
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
K?hler Geometry on Toric Manifolds, and some other Manifolds with Large Symmetry
Simon K. Donaldson
Introduction
1 Background
1.1 Gauge theory and holomorphic bundles
1.2 Symplectic and complex structures
1.3 The equations
2 Toric manifolds
2.1 Local differential geometry
2.2 The global structure
2.3 Algebraic metrics and asymptotics
2.4 Extremal metrics on toric varieties
3 Toric Fano manifolds
3.1 The K?hler-Ricci soliton equation
3.2 Continuity method, convexity and a fundamental inequality
3.3 A priori estimate
3.4 The method of Wang and Zhu
4 Variants of toric differential geometry
4.1 Multiplicity-free manifolds
4.2 Manifolds with a dense orbit
5 The Mukai-Umemura manifold and its deformations
5.1 Mukai's construction
5.2 Topological and symplectic picture
5.3 Deformations
5.4 The α-invariant
References
Gluing Constructions of Special Lagrangian Cones
Mark Haskins, Nikolaos Kapouleas
1 Introduction
2 Special Lagrangian cones and special Legendrian submanifolds of □2n-1
3 Cohomogeneity one special Legendrian submanifolds of □2n-1.
4 Construction of the initial almost special Legendrian submanifolds
5 The symmetry group and the general framework for correcting the initial surfaces
6 The linearized equation
7 Using the Geometric Principle to prescribe the extended substitute kernel
8 The main results
A Symmetries and quadratics
References
Harmonic Mappings
Jürgen Jost
1 Introduction
2 Harmonic mappings from the perspective of Riemannian geometry.
2.1 Harmonic mappings between Riemannian manifolds: definitions and properties
2.2 The heat flow and harmonic mappings into nonpositively curved manifolds
2.3 Harmonic mappings into convex regions and applications to the Bernstein problem
3 Harmonic mappings from the perspective of abstract analysis and convexity theory
3.1 Existence
3.2 Regularity
3.3 Uniqueness and some applications
4 Harmonic mappings in K?hler and algebraic geometry
4.1 Rigidity and superrigidity
4.2 Harmonic maps and group representations
4.3 K?hler groups
4.4 Quasiprojective varieties and harmonic mappings of infinite energy
5 Harmonic mappings and Riemann surfaces
5.1 Families of Riemann surfaces
5.2 Harmonic mappings from Riemann surfaces
References
Harmonic Functions on Complete Riemannian Manifolds
Peter Li
Introduction
1 Gradient estimates
2 Green's function and parabolicity
3 Heat kernel estimates and mean value inequality
4 Harmonic functions and ends
5 Stability of minimal
6 Polynomial growth harmonic functions
7 Massive sets and the structure of harmonic maps
8 Lq Harmonic functions
References
Complexity of Solutions of Partial Differential Equations
Fang Hua Lin
1 Introduction
2 Level and critical point sets
3 Solutions of nonlinear equations
4 A partition problem for eigenvalues
4.1 Heat flow for eigenfunctions
4.2 Gradient flow approach to (P)
Acknowledgement
References
Variational Principles on Triangulated Surfaces
Feng Luo
1 Introduction
2 The Schlaefli formula and its counterparts in dimension 2
2.1 Regge calculus and Casson's approach in dimension 3
2.2 The work of Colin de Verdiere, Rivin, Cohen-Kenyon-Propp and Leibon
2.3 The Cosine Law and 2-dimensional Schlaefli formulas
2.4 The geometric meaning of some action functionalsno
3 Variational principles on surfaces
3.1 Colin de Verdiere's proof of Thurston-Andreev's rigidity theorem
3.2 The work of Rivin and Leibon
3.3 New curvatures for polyhedral metrics and some rigidity theorems
3.4 Application to Teichmiiller theory of surfaces with boundary
4 The moduli spaces of polyhedral metrics
4.1 Thurston-Andreev's theorem and Marden-Rodin's proof
4.2 Some other results on the space of curvatures
4.3 A sketch of the proof theorems 3.6 and 3.7
5 Several open problems
References
Asymptotic Structures in the Geometry of Stability and Extremal Metrics Toshiki Mabuchi
1 Extremal metrics in K?hler geometry
2 Stability for polarized algebraic manifolds
3 The asymptotic Bergman kernel
4 Test configurations
5 Affine sphere equations
6 "Affine spheres" for toric Einstein surfaces
7 Asymptotic expansion for toric Einstein surfaces
References
Stable Constant Mean Curvature Surfaces
William H. Meeks III, Joaquín Pérez, Antonio Ros
1 Introduction
2 Stability of minimal and constant mean curvature surfaces
2.1 The operator ?+q
2.2 Stable H-surfaces
2.3 Global theorems for stable H-surfaces
3 Weak H-laminations
4 The Stable Limit Leaf Theorem
5 Foliations by constant mean curvature surfaces
5.1 Curvature estimates and sharp mean curvature bounds for CMC foliations .
5.2 Codimension one CMC foliations of □4 and □5
6 Removable singularities and local pictures
6.1 Structure theorems for singular CMC foliations
6.2 The Local Picture Theorem on the scale of topology
6.3 The statement of the theorem
7 Compactness of finite total curvature surfaces
7.1 The moduli space MC and the proof of Theorem 7.2
8 Singular minimal laminations
9 The moduli space of embedded minimal surfaces of fixed genus
9.1 Conjectures on stable CMC surfaces in homogeneous three-manifolds
10 Appendix
References
A General Asymptotic Decay Lemma for Elliptic Problems
Leon Simon
Introduction
1 Scale invariant compact classes of submanifolds
2 Some preliminaries concerning the class P
3 Stability inequality
4 Compact classes of cones
5 A partial Harnack theory
6 Proof of Theorem 1
7 Application to growth estimates for exterior solutions .
References
Uniformization of Open Nonnegatively Curved K?hler Manifolds in Higher Dimensions
Luen-Fai Tam
1 Introduction
2 Function theory on K?hler manifolds
2.1 Preliminary
2.2 A Liouville theorem for pluri-subharmonic functions
2.3 Polynomial growth holomorphic functions
3 Busemann function and the structure of nonnegatively curved K?hler manifolds
3.1 Curvature decay and volume growth
3.2 Manifolds with nonnegative sectional curvature
4 K?hler-Ricci flow
5 Uniformization results
5.1 Uniformization of gradient K?hler-Ricci solitons
5.2 Quadratic curvature decay
5.3 Linear curvature decay
References
Geometry of Measures: Harmonic Analysis Meets Geometric Measure Theory Tatiana. Toro
1 Introduction
2 Density-an indicator of regularity
3 Harmonic measure: boundary structure and size
4 Geometric measure theory tools
5 Open questions
References
The Monge-Ampère Eequation and its Geometric Aapplications
Neil S. Trudinger, Xu-Jia Wang
1 Introduction
2 The Monge-Ampère measure
2.1 Locally convex hypersurfaces
2.2 The Monge-Ampère measure
2.3 Generalized solutions
3 A priori estimates
3.1 Minimum ellipsoid
3.2 Uniform and H?lder estimates
3.3 Strict convexity and C1,α regularity
3.4 Second derivative estimate
3.5 C2,α estimate
3.6 W2,p estimate
3.7 H?lder estimate for the linearized Monge-Ampère equation
3.8 Monge-Ampère equations of general form
4 Existence and uniqueness of solutions
4.1 The Dirichlet problem
4.2 Other boundary value problems
4.3 Entire solutions
4.4 Hypersurfaces of prescribed Gauss curvature
4.5 Variational problems for the Monge-Ampère equation
4.6 Application to the isoperimetric inequality
5 The affine metric
5.1 Affine completeness
5.2 Affine spheres
6 Affine maximal surfaces
6.1 The affine maximal surface equation
6.2 A priori estimates
6.3 The affine Bernstein problem
6.4 The first boundary value problem
6.5 The second boundary value problem
6.6 The affine Plateau problem
References
Lectures on Mean Curvature Flows in Higher Codimensions
Mu-Tao Wang
1 Basic materials
1.1 Connections, curvature, and the Laplacian
1.2 Immersed submanifolds and the second fundamental forms
1.3 First variation formula
2 Mean curvature flow
2.1 The equation
2.2 Finite time singularity
3 Blow-up analysis
3.1 Backward heat kernel and monotonicity formula
3.2 Synopsis of singularities
4 Applications to deformations of symplectomorphisms of Riemann surfaces
4.1 Introduction
4.2 Derivation of evolution equations
4.3 Long time existence
4.4 Smooth convergence as t→∞
4.5 c＞0 case
4.6 c=0 case
4.7 c＜0 case
5 Acknowledgement
References
Local and Global Analysis of Eigenfunctions on Riemannian Manifolds
Steve Zelditch
Introduction
1 Basic definitions and notations
1.1 Planck's constant and eigenvalue asymptotics
1.2 Spectral kernels
1.3 Geodesic flow
1.4 Closed geodesics
1.5 Jacobi fields and linear Poincaré map along a closed geodesic
1.6 Geodesic flow as a unitary operator
1.7 Spectrum and geodesic flow
1.8 Ergodic, weak mixing and Anosov geodesic flows
1.9 Completely integrable geodesic flow
1.10 Quantum mechanics: wave group and pseudo-differential operators
1.11 Modes and quasi-modes
1.12 Heuristics and intutions
1.13 Notational Index
2 Explicitly solvable eigenfunctions
2
2.2 Flat tori
2.3 Standard Sphere
2.4 Surface of revolution
2.5 Hn
2.6 The Euclidean unit disc D
2.7 An ellipse
3Local behavior of eigenfunctions
3.1 Eigenfuntions and harmonic functions on a cone
3.2 Frequency function
3.3 Doubling estimate, vanishing order estimate and lower bound estimate
3.4 Semi-classical Lacunas
3.5 Three ball inequalities and propagation of smallness
3.6 Bernstein inequalities
3.7 Carleman inequalities
3.8 Geometric comparision inequalities
3.9 Symmetry of positive and negative sets
3.10 Alexandroff-Bakelman-Pucci-Cabré inequality
3.11 Bers scaling near zeros
3.12 Heuristic scaling at non-zero points
4 Nodal sets on C∞ Riemannian manifolds
4.1 Courant and Pleijel bounds on nodal domains
4.2 Critical and singular sets of eigenfunctions on C∞ Riemannian manifolds
5 The wave kernel of a compact Riemannian manifold
5.1 Manifolds without conjugate points
6 Methods for global analysis
6.1 Egorov's theorem
6.2 Sharp Garding inequality
6.3 Operator norm and symbol norm
6.4 Quantum Limits (Microlocal defect measures)
7 Singularities pre-trace formulae
7.1 Duistermaat-Guillemin short time pre-trace formula
7.2 Long time pre-trace formulae
7.3 Safarov trace formula
8 Weyl law and local Weyl law
9 Local and global Lp estimates of eigenfunctions
9.1 Sketch of proof of the Sogge Lp estimate
9.2 Generic non-sharpness of Sogge estimates
10 Gaussian beams and quasi-modes associated to stable closed geodesics
10.1 Local model
10.2 WKB ansatz for a Gaussian beam
10.3 Quantum Birkhoff normal form: intertwining to the model
11 Birkhoff normal forms around closed geodesics
11.1 Local quantum Birkhoff normal forms
11.2 Model eigenfunctions around closed geodesics
12 Quantum integrable Laplacian
12.1 Quantum integrability and ladders of eigenfunctions
12.2 Geometric examples
12.3 Localization of integrable eigenfunctions
12.4 Conjugation to normal form around torus orbits
13 Concentration and non-concentration for general (M, g)
13.1 Lp norms for restrictions to submanifolds
13.2 Non-concentration in tubes around hyperbolic closed geodesics
13.3 Non-concentration around closed geodesics on compact hyperbolic surfaces
14 Lp norms and concentration in the Quantum integrable case
14.1 Mass concentration on small length scales .
15 Delocalization in quantum ergodic systems, I
15.1 Quantum ergodicity in terms of operator time and space averages
15.2 Quantum unique ergodicity and converse quantum ergodicity
15.3 Quantum weak mixing
15.4 Spectral measures and matrix elements
15.5 Rate of quantum ergodicity and mixing
15.6 Quantum chaos conjectures
15.7 Rigorous results
15.8 Quantum limits on a hyperbolic surface and Patterson-Sullivan distributions
16 Delocalization of eigenfunctions: II: Entropy of quantum limits on manifolds with Anosov geodesic flow
17 Real analytic manifolds and their complexifications
17.1 Analytic continuation of eigenfunctions
17.2 Maximal plurisubharmonic functions and growth of φ□ λ
17.3 Analytic continuation and nodal hypersurfaces
17.4 Nodal hypersurfaces in the case of ergodic geodesic flow
17.5 Analytic domains with boundary
18 Riemannian random waves
18.1 Levy concentration of measure
18.2 Concentration of measure and Lp norms
18.3 L∞ norms: Proof of Theorem 18.3
18.4 Sup norms on small balls
18.5 Relation to Levy concentration
18.6 Nazarov-Sodin Theorem on the mean number of nodal domains of random spherical harmonics
19 Appendix on Tauberian Theorems
References
Yau's Form of Schwarz Lemma and Arakelov Inequality On Moduli Spaces of Projective Manifolds
Kang Zuo
Introduction
1 Polarized complex variation of hodge structure and Higgs bundle
2 Viehweg's positivity theorem for direct image sheaves
3 Coverings, constructing Higgs bundles and the positivity on moduli spaces
3.1 Iterated Higgs
3.2 Constructing of VHS
4 Algebraic hyperbolicity and effective boundedness
References
