A smörgåsbord of scalar-flat Kähler ALE surfaces

There are many known examples of scalar-flat Kähler ALE surfaces, all of which have group at infinity either cyclic or contained in {{\rm{SU}}(2)} . The main result in this paper shows that for any non-cyclic finite subgroup Γ \subset U(2) containing no complex reflections, there exist scalar-flat Kähler ALE metrics on the minimal resolution of \mathbb{C}^{2} /Γ, for which Γ occurs as the group at infinity. Furthermore, we show that these metrics admit a holomorphic isometric circle action. It is also shown that there exist scalar-flat Kähler ALE metrics with respect to some small deformations of complex structure of the minimal resolution. Lastly, we show the existence of extremal Kähler metrics admitting holomorphic isometric circle actions in certain Kähler classes on the complex analytic compactifications of the minimal resolutions.

A criterion for the properness of the K-energy in a general Kähler class (II)

In this paper, we give a result on the properness of the [Formula: see text]-energy, which answers a question of Song–Weinkove [On the convergence and singularities of the [Formula: see text]-flow with applications to the Mabuchi energy, Comm. Pure Appl. Math. 61(2) (2008) 210–229] in any dimensions. Moreover, we extend our previous result on the properness of [Formula: see text]-energy in [H. Li, Y. Shi and Y. Yao, A criterion for the properness of the [Formula: see text]-energy in a general Kähler class, Math. Ann. 361(1–2) (2015) 135–156] to the case of extremal [Formula: see text]-energy associated to extremal Kähler metrics.