Surveys in Geometric Analysis - Volume 3 2019.

Format:

Book

Author/Creator:

Tian, Gang.

Contributor:

Han, Qing.

Zhang, Zhenlei.

Series:

Surveys in Geometric Analysis Series

Language:

English

Physical Description:

1 online resource (268 pages)

Edition:

1st ed.

Place of Publication:

Les Ulis : EDP Sciences, 2024.

Summary:

Geometric analysis has been a very important and active research field at the frontier of mathematics for last few decades, and many important achievements have been made in this field. The development of geometric analysis has played a key role in establishing numerous basic theories and solving long-standing problems in mathematics. It is a field worthy of much continued exploration.Since the late 1990s, Peking University has taken the lead in organizing the annual Workshop on Geometric Analysis with participation of universities and research institutions from China and abroad. This work refers to the 2019 edition of the Workshop and is the third volume of a dedicated book series that aims to present the progress of Chinese and global research in geometric analysis by curating a selection of high-quality papers submitted by scholars to the annual Workshop on Geometric Analysis. It provides readers with a comprehensive overview of the latest trends, methods, and breakthroughs in the field, and serves as a valuable resource for ongoing research.

Contents:

Cover-1

Cover-2

Surveys in Geometric Analysis - Volume 3, 2019

Prologue

Contents

Complete Noncompact Kahler Manifolds with Positive Curvature

1 Introduction

2 Geometric properties

3 Maximal volume growth

4 Non-maximal volume growth

References

Extreme Gap Problems in Random Matrix Theory

1 Background

2 Smallest gaps

3 Largest gaps

Classification on Singularity Types of Long-Time Kahler-Ricci Flow

1.1 Backgrounds

1.2 Long-time Kahler-Ricci flow

1.3 Semi-ample canonical line bundle

1.4 Main interest of this paper: infinite-time singularity type

2 Early and recent classification results

2.1 Early results

2.2 Recent results

2.3 Case (*)

3 Progresses on Case (*)

3.1 Independence on the initial metric

3.2 Local singularity types

3.3 Criterions for type IIb solutions

3.3.1 Singularity type and characteristic index

3.3.2 Singularity type and existence of certain holomorphic maps

3.4 Final remark

Scalar Curvatures on an Almost Hermitian Manifold

2 Applications

3 A sketch of the proof for the main result

3.1 To compute s-sJ

3.2 To compute s1(0)-1/2SJ and s2(0)-1/4(s+sJ)

3.3 To compute s1(t)-s1(0) and s2(t)-s2(0)

4 Asides

4.1 On coformal flat metrics

4.2 On compact Hermitian metrics with s1(-1) = s2(-1)

The Kahler-Ricci Flow with Log Canonical Singularities

2 A brief introduction to minimal model Program (MMP)

3 Kahler-Ricci flow and analytic minimal model program (AMMP)

4 Main results

5 The strategy of the proof

6 Further discussions

Problems Related to Isoparametric Theory

2 Problems related to isoparametric theory on unit spheres

3 Problems related to isoparametric theory generally.

References

Some Recent Results on the Lp-Minkowski Problem

1.1 Lp-Minkowski problem

1.2 Dual Lp-Minkowski problem

2 Uniqueness and non-uniquenes results

2.1 Lutwak-Yang-Zhang conjecture

2.2 Non-uniqueness results

3 Noncompact case

3.1 Existence results

3.2 Other remarks

3.2.1 Homogeneous extension

3.2.2 Non-existence

3.2.3 Gauss curvature flow

3.2.4 Noncompact dual Lp-Minkowski problems

The Lp-Minkowski Problem

2 Self-similar solution to Gauss curvature flows

3 Known results and open problems

High-Order Regularity of Collapsing Metrics in Kahler Geometry

2 High-order estimates of collapsing Calabi-Yau metrics of Hein-Tosatti

3 A "Boundedness Implies Convergence" principle and its applications to collapsing Calabi-Yau metrics

4 Applications to normalized Kahler-Ricci flow on torus-fibered minimal models

Optimal Boundary Regularity for Fast Diffusion Equations

2 Two main steps in our proof

2.1 Linear theory

2.2 A priori estimates

Sphere Theorems for Submanifolds in Kahler Manifold

2 Preliminaries

3 Some algebraic estimates

4 Proof of the theorems

5 Sphere theorems for lagrangian submanifolds

On the Existence of the Mabuchi Soliton

2 Preliminary

3 The continuity method

4 Futaki invariant on Q-Fano varieties

5 Main Result

Rigidity of Entire Self-Shrinking Solutions to Lagrangian Mean Curvature Flow

2 Rigidity results

3 Strategy of proof

4 Complex counterparts

4.1 Entire Kahler-Ricci soliton

4.2 Entire self-similar solutions to deformed Hermitian-Yang-Mills equation

4.3 Entire self-similar solutions to J-flow

References.

Morse Theory for Minimal Hypersurfaces and the Multiplicity One Conjecture

2 Morse theory for the area functional

3 Multiplicity One Conjecture

4 H-hypersurfaces

5 Approaching by sweepouts of boundaries

6 Multiplicity one for free boundary minimal hypersurfaces and applications

Type-II Solutions to Mean Curvature Flow

2 Level set flow

3 Type-II examples

4 Local analysis for symmetric type-II solutions

The Greatest Ricci Lower Bound and Stability Threshold

1 Motivation

2 Background

3 Main result

4 The proof

The Hermitian-Einstein Equation on Higgs Bundle and Its Applications

2 A generalized D-U-Y theorem on Higgs bundle

3 Limiting behavior of the Hermitian-Yang-Mills flow

4 A filtration of semi-stable Higgs bundle

5 Stable Higgs bundle and irreducible flat bundle

6 Semi-stable Higgs bundle and flat bundle

Existence of Solutions to Mean Field Equations on Closed Riemann Surfaces

2 Existence of solutions to mean field equations

3 A flow method to solve the mean field equations at noncritical cases

4 The convergence of the mean field type flow at a critical case

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9782759836642

2759836649

OCLC:

1453270618