Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one —IV

Abstract

This paper is the first of a series of papers in which we generalize our results in (Asian

Affine compact almost-homogeneous manifolds of cohomogeneity one

“…But JF α = −G α (mod ) as it is in the tangent space of G/P. Therefore, we have JF α = −G α for any α which is not in the root system of S. This discussion is corresponding to the discussion in the last paragraph of the second section of [15].…”

“…Therefore, we are only interested in those manifolds which are almost-homogenoues of cohomogeneity one with A 0 (M) action. We have (see [14, section 12] In [15,18] and this paper, we consider the simply connected affine and the type II cases with a hypersurface end. In [17], we shall consider the simply connected type I cases with a hypersurface end.…”

“…This of course should be true, but we just use it as examples. In this paper, as in [15] we also have three natural variables: the nipotent time, θ the phase angle, τ the micro time. They help us understand the equation in detail.…”

“…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. By taking advantage of the solution for higher codimensional ends in [9], we also checked the possibility of blowing down of our manifolds.…”

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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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