Existence of Extremal Metrics on Almost Homogeneous Manifolds of Cohomogeneity One — Iii

Abstract

In this paper we prove that on certain manifolds Nn with nonnegative first Chern class the existence of extremal metric in a Kähler class is the same as the stability of the Kähler class. We also obtain many new Kähler classes with extremal metrics, in particular, the Kähler-Einstein metrics for Nn with n > 2. We also compare the problem of finding Calabi extremal metrics with the similar problem of finding Hermitian-Einstein metrics on the holomorphic vector bundles. We explain the geodesic stability and found that the stability for the manifold is much more complicated

Extremal solitons and exponential C∞convergence of the modified Calabi flow on certain ℂP1Bundles

“…Our results can be regarded as a continuation of [Koiso and Sakane 1986;1988;Koiso 1990;Guan 1993;1995a;1995b;1999;2003]. For the reader unfamiliar with those papers we state, without detailed proof, several lemmas and Theorem 2.10 below, which mostly can be found in [Guan 1995a] (Lemmas 2.2 and 2.3 are from [Guan 1999]).…”

“…In [Guan 2003] we showed that the existence of this unique extremal metric is equivalent to the geodesic stability of the Kähler class. It is natural to ask:…”

Affine compact almost-homogeneous manifolds of cohomogeneity one

“…We say that a manifold is of cohomogeneity one if the maximal compact subgroup has a (real) hypersurface orbit. In [20] and [14], we reduced compact complex almost homogeneous manifolds of cohomogeneity one into three types of manifolds. We denote the manifold by M and let G be a complex subgroup of its automorphism group which has an open orbit on M. Let us assume first that M is simply connected.…”

“…The choice θ = 2 + 2 makes the equation much simpler. We avoided another natural variable the semisimple time which was in [14], but it will eventually appear in [19]. As in [15], the energy norm function and the Ricci mixed energy norm function ρ in the sections 4 and 6 are seemly God given, which are the reasons that we can solve this probem.…”

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Abstract: This paper is one in a series generalizing our results in [12,14,15,20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends.

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