Existence of Extremal Metrics on Compact Almost homogeneous Kahler Manifolds with Two Ends

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

“…Hence we may apply the method of counting zeros in [9,11] to f l . The quantity x x (8 − x) is proportional to ϕ Q in [9].…”

“…In 1992 [8,9] we proved the existence of Calabi extremal metrics for complex compact almost homogeneous manifolds with two ends. We were able to reduce a fourth order ordinary equation to two first order equations.…”

Type II almost-homogeneous manifolds of cohomogeneity one

“…We observe that g l = 2v f l is actually a polynomial in v and should be proportional to the derivative of "ϕ Q" in [Guan 1995]; thus we expect that y = 2 l (−B − v) − 1 corresponds to the "U " in that paper. We let q = 2v Q(v), and observe that q is proportional to the "Q" in [Guan 1995]. We have…”

“…As in [Guan 2002], we may expect that x is related to a Kähler metric of constant scalar curvature on the normal line bundle over the hypersurface orbit. Hence, we may apply the method of counting zeros in [Guan 1995;2002] to this circumstance. The expression x n−1 x Q 1 (x) is proportional to "ϕ Q" in the first of these papers.…”

“…Type III compact complex almost-homogeneous manifolds of real cohomogeneity one were dealt with in [Guan 1995]. Not much stability is found there (but see [Guan 2003] for the stability of related constructions).…”