Surveys in Geometric Analysis - Volume 4 2020.

Format:

Book

Author/Creator:

Tian, Gang.

Contributor:

Han, Qing.

Zhang, Zhenlei.

Series:

Surveys in Geometric Analysis Series

Language:

English

Physical Description:

1 online resource (140 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 2020 edition of the Workshop and is the fourth 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 4, 2020

Prologue

Contents

Conformal Metrics of Constant Scalar Curvature and Constant Boundary Mean Curvature

1 Motivation

2 Preliminary

3 Constrained variational problems

3.1 Existence of minimizers: Subcritical approximations

3.2 Curvature flow approach

4 Han-Li's conjecture: Mountain Pass critical points

5 Open problems

5.1 Geometric rigidity

5.2 Remaining case in Han-Li's conjecture

References

Some Schwarz Type Lemmas on Pseudo-Hermitian Manifolds

1 Introduction

2 Pseudo-Hermitian geometry

3 Schwarz type lemmas on pseudo-Hermitian manifolds

3.1 CR maps between psuedo-Hermitian manifolds

3.2 Transversally holomorphic maps between pseudo-Hermitian manifolds

3.3 (J, JN)- and (JN, J)- holomorphic maps

Curvature Flows in Hyperbolic Space and Their Applications

1.1 Huisken's proof of isoperimetric inequality

1.2 Guan-Li's proof of quermassintegral inequalities

2 Preliminaries

2.1 Elementary symmetric functions

2.2 Hypersurface geometry in hyperbolic space

2.3 Quermassintegrals for convex domains in hyperbolic space

3 Geometric inequalities

3.1 Inverse curvature flows

3.2 Globally constrained curvature flows

3.3 Contracting curvature flows

3.4 Locally constrained curvature flows

Localization of η-Invariants and Differential K-Theory

2 Atiyah-Segal localization formula

3 Localization of η-invariants

4. Differential K-theory

5 Comparison of equivariant η-invariants

A Brief Survey on Gromov-Hausdorff Convergence of Kahler Manifolds

2 Riemannian case

3 Kahler case

The Relative Isoperimetric Inequalilty

2 A generalized cone and its area estimate.

3 A new proof of the relative isoperimetric inequality

4 Relative isoperimetric inequality for minimal submanifolds

5 A logarithmic Sobolev inequality for a free boundary submanifold

An Eigenvalue Estimate for the ∂-Laplacian Associated to a Line Bundle with Singular Metrics

2 Applications

3 The modified ideal sheaf

4 A sketch of the proof for the main result

4.1 Siu's ∂∂-Bochner formula

4.2 The submeanvalue estimate: from X to ∣r∣ &lt

1

4.3 The submeanvalue estimate: from ∣r∣ &lt

1 to one point

4.4 The proof of Theorem 1.1

Stability Thresholds and Canonical Metrics

1 Motivation: Kahler-Einstein problem

2 Alpha invariant

2.1 Analytic definition

2.2 Quantized alpha invariant

2.3 Algebraic definition

2.4 Valuative characterization of alpha invariant

2.5 Recent results on α-invariants

3 Delta invariant

3.1 Valuative characterization of δ-invariant

3.2 Delta invariant with higher momentum

3.3 Bounding the volume of Fano manifolds using delta invariant

3.4 Computation of δ-invariant

3.5 The greatest Ricci lower bound

3.6 Analytic delta invariant

4 Quantized delta invariant and balanced metrics

4.1 Qunatized Monge-Ampere energy

4.2 δAm-invariant

4.3 The identity δAm = δm

4.4 Quantized Ding energy and balanced metrics

References.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

9782759836581

2759836584

OCLC:

1455118187