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By studying a complex Monge-Amp`ere equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact Kahler manifold $N^n$ with $Ric_k< 0$ for some integer $k$ with $1<k<n$, and the ampleness of the canonical line bundle $K_N$.
The purpose of this paper is to establish a Lagrangian potential theory, analogous to the classical pluripotential theory, and to define and study a Lagrangian differential operator of Monge-Ampere type. This development is new even in ${bf C}^n$. Ho
We obtain a genuine local $C^2$ estimate for the Monge-Amp`ere equation in dimension two, by using the partial Legendre transform.
A new proof for stability estimates for the complex Monge-Amp`ere and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations o
In this paper we further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities and new affine invariant entropy inequalities. We obtain new inequalities on functional affine su
In this paper, the author studies quaternionic Monge-Amp`ere equations and obtains the existence and uniqueness of the solutions to the Dirichlet problem for such equations without any restriction on domains. Our paper not only answers to the open pr