Extremal metrics on ruled manifolds

Minimizing Weak Solutions for Calabi’s Extremal Metrics on Toric Manifolds

The existence of extremal metrics has been recently studied extensively on Kähler manifolds. The goal is to establish a sufficient and necessary condition for the existence of extremal metrics in the sense of Geometric Invariant Theory. There are many important works related to the necessary part ([Ti], [D1], [M1],[M2]). The sufficient part seems more difficult than the necessary part since the...

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.

Extremal Metrics on Graphs I

We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants (girth, diameter), and spectral invariants (the determinant of the Laplace operator, or complexity; bottom non-zero eigenvalue of the Laplace operator). We show ...

Existence of Extremal Metrics on Almost Homogeneous Manifolds of Cohomogeneity One∗

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

This paper is the first of a series of papers in which we generalize our results in (Asian J. of Math. 4, 817–830 (2000); J. Geom. Anal. 12, 63–79 (2002); Intern. J. Math. 14, 259–287 (2003)) to the general complex compact almost homogeneous manifolds of real cohomogeneity one. In this paper we deal with the exceptional case of the G2 action (Cf. Intern. J. Math. 14, 259–287 (2003), p. 285). In...