On the constant scalar curvature Kähler metrics, existence results

Abstract

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of Kähler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $({\mathcal{E}}^1,d_1)$ are smooth.

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Cited by in corpus (26)

• Existence of twisted constant scalar curvature Kähler metrics with a large twist
• Optimal lower bounds for Donaldson's J-functional
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• Nakai-Moishezon criterions for complex Hessian equations
• The closures of test configurations and algebraic singularity types
• Continuity of delta invariants and twisted Kähler--Einstein metrics
• Existence of cscK metrics on smooth minimal models
• New curvature flows in complex geometry
• Scalar curvature and properness on Sasaki manifolds
• The Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds
• Geometrical pluripotential theory on Sasaki manifolds
• A remark on constant scalar curvature Kähler metrics on minimal models
• On Calabi's extremal metric and properness
• Conic singularities metrics with prescribed scalar curvature: a priori estimates for normal crossing divisors
• Tian's properness conjectures: an introduction to Kahler geometry
• On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature
• Geodesics in the space of $m$-subharmonic functions with bounded energy
• Openness of uniform K-stability in the Kähler cone
• Mahler measures, Stable Pairs, and the Global coercive estimate for the Mabuchi Functional
• Local curvature estimates along the $\kappa$-LYZ flow
• A uniform version of the Yau-Tian-Donaldson correspondence for extremal Kähler metrics on polarized toric manifolds
• Faltings Heights, Igusa Local Zeta Functions, and the Stability Conjectures in Kahler Geometry I
• Convergence of cscK metrics on smooth minimal models of general type
• Rotation invariant singular Kähler metrics with constant scalar curvature on ${\mathbb{C}}^n$
• L^1 metric geometry of big cohomology classes