Canonical Metrics in Kähler Geometry

2000
ISBN: 978-3-7643-6194-5
Verlag: Springer

Buch, Englisch, 101 Seiten, Paperback, Format (B × H): 178 mm x 254 mm, Gewicht: 228 g

Reihe: Lectures in Mathematics. ETH Zürich

There has been fundamental progress in complex differential geometry in the last two decades. For one, The uniformization theory of canonical Kähler metrics has been established in higher dimensions, and many applications have been found, including the use of Calabi-Yau spaces in superstring theory. This monograph gives an introduction to the theory of canonical Kähler metrics on complex manifolds. It also presents some advanced topics not easily found elsewhere.

Zielgruppe

Research

Autoren/Hrsg.

Tian, Gang

Weitere Mitwirkende

Akveld, M.

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Naturwissenschaften Biowissenschaften Angewandte Biologie Biomathematik
• Mathematik | Informatik Mathematik Geometrie Nicht-Euklidische Geometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

1 Introduction to Kähler manifolds.- 1.1 Kähler metrics.- 1.2 Curvature of Kähler metrics.- 2 Extremal Kähler metrics.- 2.1 The space of Kähler metrics.- 2.2 A brief review of Chern classes.- 2.3 Uniformization of Kähler-Einstein manifolds.- 3 Calabi-Futaki invariants.- 3.1 Definition of Calabi-Futaki invariants.- 3.2 Localization formula for Calabi-Futaki invariants.- 4 Scalar curvature as a moment map.- 5 Kähler-Einstein metrics with non-positive scalar curvature.- 5.1 The Calabi-Yau Theorem.- 5.2 Kähler-Einstein metrics for manifolds with c1(M) < 0.- 6 Kähler-Einstein metrics with positive scalar curvature.- 6.1 A variational approach.- 6.2 Existence of Kähler-Einstein metrics.- 6.3 Examples.- 7 Applications and generalizations.- 7.1 A manifold without Kähler-Einstein metric.- 7.2 K-energy and metrics of constant scalar curvature.- 7.3 Relation to stability.