On the constant scalar curvature Kähler metrics, general automorphism group

Abstract

In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when $Au{t}_0(M,J)\ne 0$. Moreover, we also show that when $Au{t}_0(M,J)\ne 0$, the properness of $K$-energy with respect to a suitably defined distance implies the existence of cscK.

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Cited by in corpus (20)

• Geodesic rays and stability in the cscK problem
• Optimal lower bounds for Donaldson's J-functional
• $L^1$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds
• Uniform K-stability of polarized spherical varieties
• Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons
• The closures of test configurations and algebraic singularity types
• Existence of cscK metrics on smooth minimal models
• Scalar curvature and properness on Sasaki manifolds
• Mabuchi geometry of big cohomology classes with prescribed singularities
• Geometrical pluripotential theory on Sasaki manifolds
• On Calabi's extremal metric and properness
• Tian's properness conjectures: an introduction to Kahler geometry
• Conic singularities metrics with prescribed scalar curvature: a priori estimates for normal crossing divisors
• Geodesics in the space of $m$-subharmonic functions with bounded energy
• Openness of uniform K-stability in the Kähler cone
• Local curvature estimates along the $\kappa$-LYZ flow
• Mahler measures, Stable Pairs, and the Global coercive estimate for the Mabuchi Functional
• A uniform version of the Yau-Tian-Donaldson correspondence for extremal Kähler metrics on polarized toric manifolds
• Rotation invariant singular Kähler metrics with constant scalar curvature on ${\mathbb{C}}^n$
• Faltings Heights, Igusa Local Zeta Functions, and the Stability Conjectures in Kahler Geometry I