A criterion for the properness of the K-energy in a general Kahler class

Abstract

In this paper, we give a criterion for the properness of the K-energy in a general Kahler class of a compact Kahler manifold by using Song-Weinkove's result. As applications, we give some Kahler classes on $\mathbb{C}{\mathbb{P}}^2\#3\overline{\mathbb{C}{\mathbb{P}}^2}$ and $\mathbb{C}{\mathbb{P}}^2\#8\overline{\mathbb{C}{\mathbb{P}}^2}$ in which the K-energy is proper. Finally, we prove Song-Weinkove's result on the existence of critical points of $\hat{J}$ functional by the continuity method.

20 pages, final version, to appear in Mathematische Annalen

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Cited by in corpus (1)

• Geodesic rays and stability in the cscK problem