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29 March 2024
 
  » arxiv » math.DG/0108162

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The Space of Kähler metrics (II)
E. Calabi ; X. X. Chen ;
Date 23 Aug 2001
Subject Differential Geometry | math.DG
AbstractThis 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 cite{chen991} 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.
Source arXiv, math.DG/0108162
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