Quasi-Einstein Metrics

Abstract

Motivated by Koiso’s work on quasi-Einstein metrics on Fano manifolds, we define (generalized) quasi-Einstein metrics in any Kähler class on any compact complex manifold. It turns out that these metrics are similar to Calabi’s Extremal metrics. Moreover their existence might be studied by a curvature flow in a given Kähler class.

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

“…Kähler Ricci-solitons were first studied by H. D. Cao [1996], Koiso [1987;1990] and Tian, which was motivated by Hamilton's similar work on Ricci-solitons in the Riemannian case. (In that direction, there are some interesting results in [Koiso 1990;Guan 1995b;Tian and Zhu 2002]. )…”

“…To interpolate extremal metrics and the quasi-Einstein metrics of [Guan 1995b], which are a kind of Kähler-soliton metrics as a generalization of Ricci-soliton metrics, we define extremal solitons. A Kähler metric is an extremal soliton if…”

“…1 One might regard the major terms as a heat equation for log(det g t / det g 0 ). That was the motivation for our considering this equation back in 1992; see [Guan 1995b]. 2 However, I could only solve the equation for metrics assuming a certain condition; the condition can be checked to hold for many concrete cases, but I could not prove it for all of our cases (see Section 11).…”

Affine compact almost-homogeneous manifolds of cohomogeneity one

“…There always exists an extremal metric in any Kähler class. Recently, we generalized this existence result to a family of metrics, which connects the extremal metric [10] and the generalized quasi-einstein metric [9], called the extremal-soliton metrics in [16]. The existence of the extremal-soliton is the same as the geodesic stability with respect to a generalized Mabuchi functional.…”

“…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. In all our calculations we also need to take care of the change of the invariant inner products when we restrict our calculation to a typical subgroup S in G.…”