Extremal K\"ahler metrics

This paper is a survey of some recent progress on the study of Calabi's extremal K\"ahler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture has been established. We then turn to the question of what one expects when no extremal metric exists.Comment: 17 pages, 4 figures. Contribution to the proceedings of the 2014 IC

Extremal metrics and K-stability

We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.Comment: 13 pages, v3: fixed typo

Blowing up extremal K\"ahler manifolds II

This is a continuation of the work of Arezzo-Pacard-Singer and the author on blowups of extremal K\"ahler manifolds. We prove the conjecture stated in [32], and we relate this result to the K-stability of blown up manifolds. As an application we prove that if a K\"ahler manifold M of dimension greater than 2 admits a cscK metric, then the blowup of M at a point admits a cscK metric if and only if it is K-stable, as long as the exceptional divisor is sufficiently small.Comment: 36 pages, fixed a citatio

Remark on the Calabi flow with bounded curvature

In this short note we prove that if the curvature tensor is uniformly bounded along the Calabi flow and the Mabuchi energy is proper, then the flow converges to a constant scalar curvature metric.Comment: 7 page