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 $Aut_0(M,J)\neq0$. Moreover, we show that when $Aut_0(M,J)\neq0$, the properness of $K$-energy with respect to a suitably defined distance implies the existence of cscK.
In this paper, we derive apriori estimates for constant scalar curvature K\"ahler metrics on a compact K\"ahler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the K\"ahler potential.
We also discuss some local versions of these estimates which can be of independent interest.
We prove uniform gradient and diameter estimates for a family of geometric
complex Monge–Ampere equations. Such estimates can be applied to study geometric regularity
of singular solutions of complex Monge–Ampere equations. We also prove a uniform
diameter estimate for collapsing families of twisted Kahler–Einstein metrics on Kahler manifolds
of nonnegative Kodaira dimensions.