Copolar convexity

K-能量的凸性与极值度量的唯一性——一个综述性介绍

Convexity of the K-energy and Uniqueness of Extremal metrics -- An Expository Introduction

本文综述了Mabuchi K-能量泛函在Kähler势空间沿弱测地线凸的证明思想，以及由此导出的常标量曲率Kähler度量和极值度量的唯一性，重点介绍概念框架与关键技术。

Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics

We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogenuous Monge-Ampere equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.

Interpolation of weighted extremal functions

An approach to complex interpolation of compact subsets of $\Bbb C^n$, including Brunn-Minkowski type inequalities for the capacities of the interpolating sets, was developed recently by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn-Minkowski inequalities.

Convexity properties related to extremal functions

In 1962 J. Siciak introduced polynomial extremal function. The Siciak extremal function is one of the most important tool to investigate derivatives of polynomials. W. Pleśniak in the circle of papers (most important joint with W. Pawłucki) showed deep connections between behavior of Siciak extremal function near compact $K$ and bounds for derivatives of polynomials. In 1990 W. Pleśniak introduced condition (P) which is equivalent to Markov property of compact $K$. In the same paper there was stated a problem which property of Siciak's extremal function are necessary to Markov's property. A st…