A note on the $K$-stability on toric manifolds

Abstract

In this note, we prove that on polarized toric manifolds the relative $K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and also we show that the modified $K$-energy is proper in the space of $G_0$-invariant Kähler metrics for the case of toric surfaces which admit the extremal metrics.

8 pages

References in corpus (1)

• An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II

Cited by in corpus (2)

• Test configurations and Geodesic rays
• Ambikaehler geometry, ambitoric surfaces and Einstein 4-orbifolds