Non-homogeneous Kähler-Einstein metrics on compact complex manifolds

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]).…”

Quaternionic reduction and Einstein manifolds

“…When M - Even dimensional examples of compact non-homogeneous Einstein manifolds with positive cosmological constant were introduced first by Page [Pa] and later his construction was generalized by Berard Bergery [BeBer]. Koiso and Sakane [KoiSal,KoiSa2] Recently Wang introduced another interesting construction of Einstein metrics on some principal bundles over products of quaternionic Kahler manifolds [Wanl]. In a sense it is a quaternionic analogue of the construction in Using our non-homogeneous examples of weakly regular 3-Sasakian structures and the theorem above one can construct many new non-homogeneous Einstein .metrics of positive scalar curvature in dimension 4A; + 3, k > 1.…”

Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends

“…In [Sk] and [KS1,2] Koiso and Sakane found a method for constructing certain Fano manifolds and proved that there exist Kähler-Einstein metrics on those Fano manifolds if and only if the Futaki invariants are zero. Recently by the same argument Andy Hwang [Hw] proved that there always exist extremal metrics on those Fano manifolds (although we finished our main theorem a little earlier, we expected that he would finish the Fano case at that time).…”

“…But the general case includes all the Hirzebruch surface F". In this paper, by considering the symmetric scalar curvature equation, a method similar to that of [KS] and [Hw] is employed to simplify the proof of [KS1] and [Hw]. The existence of extremal metrics in a more general case is proved, and the results in [KS1] and [Hw] [Gul]), which enables us to solve the general case.…”

“…In this paper, by considering the symmetric scalar curvature equation, a method similar to that of [KS] and [Hw] is employed to simplify the proof of [KS1] and [Hw]. The existence of extremal metrics in a more general case is proved, and the results in [KS1] and [Hw] [Gul]), which enables us to solve the general case. The calculation of the scalar curvature in Lemma 2 and the observation in Lemma 3, as well as the proof of Lemma 7, are essential for our proof.…”