Surveys in Geometric Analysis - Volume 2 2018.

Format:

Book

Author/Creator:

Tian, Gang.

Contributor:

Han, Qing.

Zhang, Zhenlei.

Series:

Surveys in Geometric Analysis Series

Language:

English

Physical Description:

1 online resource (186 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 2018 edition of the Workshop and is the second 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 2, 2018

Prologue

Contents

Fundamental Gap Comparison

1 Introduction

1.1 Neumann boundary/Closed manifolds

1.2 Dirichlet Boundary Condition

1.3 Further Questions

2 Basic Approach and Modulus of Continuity

3 Back to Super Log-Concavity

References

Combinational Ricci Flows with Applications to 3-dimensional Geometry and Topology

0 Introduction

1 Circle Pattern on Surfaces of Finite Topological Type

1.1 Background

1.2 Main results

2 Combinational Ricci Flows with Ideal Circle Patterns

2.1 Ideal circle patterns

2.2 Hyperbolic geometry background

2.3 Euclidean geometry background

2.4 An application to Rivin's theorem

3 Rigidity of the Hexagonal Delaunay Triangulated Plane

3.1 Background

3.2 Rigidity of the hexagonal Delaunay triangulated plane

3.3 PL conformal vs. hyperbolic geometry

3.4 Delaunay triangulations and convex hyperbolic polyhedra

3.5 A hyperbolic geometry interpretation of Theorem 3.4

ϵ-Regularity of Harmonic-Einstein Equations

2 ϵ-Regularity Theorem on Einstein Manifolds

3 ϵ-Regularity Theorem with Critical Metrics

4 ϵ-Regularity Theorem with Harmonic-Einstein Equations

Free Boundary Minimal Hypersurfaces with Least Area

2 Basic Notations

3 Idea of Theorem 2

3.1 Some examples

3.2 Idea of the Proof

Babuska Problem in Composite Material

1 Motivation and Babuska Problem

2 A Simplified Model: the Conductivity Problem

3 The Elasticity Case and Optimal Blow-up Rates

4 The Convexity and the Blow-up Rates

Energy Functionals and Canonical Mertics on Polarized Compactification of Lie Groups

2 Polarized Compactifications.

3 The K-energy on General Polarized Compactifications

4 The Case of Fano Compactifications

4.1 Modified K-energy and Kahler-Ricci solitons

4.2 Modified Ding functional and Mabuchi metrics

5 Low Rank Examples

Clifford Algebra, Isoparametric Foliation and Related Geometric Constructions

2 Generalizations of OT-FKM Construction

3 Pinkall-Thorbergsson Consruction

3.1 Extrinsic geometry

3.2 Scalar curvature

3.3 Mean curvature flow

The Kahler-Ricci Flow with Cusp Singularities on Projective Manifolds

2 Carlson-Griffiths Form

3 Standard and Superstandard Spatial Asymptotics

4 A Simultaneously Complete Construction

5 General Case: Currents with Zero Lelong Number

6 Further Discussions

On the Rigidity and Gap Theorem for Liouville's Equation

1 Background

2 The Polyhomogeneous Expansions

3 Examples

4 Main Results

5 The Strategy of the Proof

Existence of Conic Kahler-Einstein Metrics and Log K-Polystable

2 Smoothing Conic Kahler-Einstein Metric

3 Compactness and Partial C0 Estimate

4 Existence of Conic Kahler-Einstein Metric

Rigidity of Free Boundary CMC Hypersurfaces in a Ball

2 Preliminaries and Jellett Type Theorem

3 Stable Free Boudary CMC Hypersurfaces

Rigidity of Entire Self-Shrinking Solutions to Kahler-Ricci Flow: An Overview

2 Rigidity results

3 Rigidity Results for Similar Equations

4 Summary and Prospect

Some Recent Results on Nirenberg Problem

1 The Classical Nirenberg Problem

2 Nirenberg Problem for Qn/2-Curvature

3 Nirenberg Problem of Fractional Orders

4 The Proofs

References.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9782759836550

275983655X

OCLC:

1453270820