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Register a new userAn application of a $C^2$-estimate for a complex Monge-Amp`ere equation
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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 $K_N$.
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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 ${bf C}^n$. However, it applies quite generally -- perhaps most importantly to symplectic manifolds equipped with a Gromov metric. The Lagrange Monge-Ampere operator is an explicit polynomial on ${rm Sym}^2(TX)$ whose principle branch defines the space of Lag-harmonics. Interestingly the operator depends only on the Laplacian and the SKEW-Hermitian part of the Hessian. The Dirichlet problem for this operator is solved in both the homogeneous and inhomogeneous cases. The homogeneous case is also solved for each of the other branches. This paper also introduces and systematically studies the notions of Lagrangian plurisubharmonic and harmonic functions, and Lagrangian convexity. An analogue of the Levi Problem is proved. In ${bf C}^n$ there is another concept, Lag-plurihamonics, which relate in several ways to the harmonics on any domain. Parallels of this Lagrangian potential theory with standard (complex) pluripotential theory are constantly emphasized.
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 of the background metric in the case of the Monge-Amp`ere equation, and under degenerations to a big class in the case of Hessian equations.
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 surface area and lower and upper bounds for the Kullback-Leibler divergence in terms of functional affine surface area. The functional inequalities lead to new inequalities for L_p-affine surface areas for convex bodies.
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 problem proposed by Semyon Alesker in [3], but also extends relevant results in [7] to the quaternionic vector space.
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