Extremal Almost-kähler Metrics

Extremal Kähler metrics and K-stability

On the Kähler Classes of Extremal Metrics

Let (M,J) be a compact complex manifold, and let E ⊂ H1,1(M,R) be the set of all cohomology classes which can be represented by Kähler forms of extremal Kähler metrics, in the sense of Calabi [3]. Then E is an open subset. ∗Supported in part by NSF grant DMS 92-04093. †Supported in part by NSF grant DMS 90-22140.

On Modified Mabuchi Functional and Mabuchi Moduli Space of Kähler Metrics on Toric Bundles

Mabuchi introduced the Mabuchi functional in [Mb1], and it turns out that it is very useful for dealing with Kähler metrics with constant scalar curvatures on compact manifolds (see [BM], etc.). One can also expect that the existence of Kähler metric with constant scalar curvature is almost equivalent to the existence of a lower bound of the Mabuchi functional (see, e.g., [Ti]). But for the cas...

Polarized 4-Manifolds, Extremal Kähler Metrics, and Seiberg-Witten Theory

Using Seiberg-Witten theory, it is shown that any Kähler metric of constant negative scalar curvature on a compact 4-manifold M minimizes the L-norm of scalar curvature among Riemannian metrics compatible with a fixed decomposition H(M) = H ⊕ H−. This implies, for example, that any such metric on a minimal ruled surface must be locally symmetric.

Asymptotic Properties of Extremal Kähler Metrics of Poincaré Type

Consider a compact Kähler manifold X with a simple normal crossing divisor D, and define Poincaré type metrics on X\D as Kähler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincaré type Kähler metric on X\D implies the existence of a constant scalar curvature (resp. an extremal) Kähler metric, possibly of Poincar...