The Space of Kähler metrics (II)

2001/08/23 by E. Calabi, Calabi, E., X. X. Chen +1 · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.math/0108162

16 pages

arxiv created 2001/08/23 · arxiv updated 2009/11/30

Abstract

This paper, the second of a series, deals with the function space of all smooth Kähler metrics in any given closed complex manifold M in a fixed cohomology class. The previous result of the second author \citechen991 showed that the space is a path length space and it is geodesically convex in the sense that any two points are joined by a unique path, which is always length minimizing and of class C^1,1. This already confirms one of Donaldson's conjecture completely and verifies another one partially. In the present paper, we show first of all, that the space is, as expected, a path length space of non-positive curvature in the sense of A. D. Alexanderov. The second result is related to the theory of extremal Kähler metrics, namely that the gradient flow of the K energy is strictly length decreasing on all paths except those induced by a path of holomorphic automorphisms of M. This result, in particular, implies that extremal Kähler metric is unique up to holomorphic transformations, provided that Donaldson's conjecture on the regularity of geodesic is true.

Cited by

• Calabi flow with bounded L^p scalar curvature (II)
• Lines in the space of Kähler metrics

Related

• The Space of Kaehler metrics
• Space of Kähler metrics (IV)--On the lower bound of the K-energy
• Space of Kähler metrics III--On the lower bound of the Calabi energy and geodesic distance
• Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics
• Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics