$L^2$ estimates of Poincaré-Lelong equations on convex domains in $\mathbb{C}^n$

$m$-potential theory and $m$-generalized Lelong numbers associated with $m$-positive supercurrents

In this study, we first define the local potential associated to a weakly positive closed supercurrent in analogy to the one investigated by Ben Messaoud and El Mir in the complex setting. Next, we study the definition and the continuity of the $m$-superHessian operator for unbounded $m$-convex functions. As an application, we generalize our previous work on Demailly-Lelong numbers and several related results in the superformalism setting. Furthermore, strongly inspired by the complex Hessian theory, we introduce the Cegrell-type classes as well as a generalization of some $m$-potential result…

Higher Lelong numbers and convex geometry

We prove the reversed Alexandrov-Fenchel inequality for mixed Monge-Ampère masses of plurisubharmonic functions, which generalizes a result of Demailly and Pham. As applications to convex geometry, this gives a complex analytic proof of the reversed Alexandrov-Fenchel inequality for mixed covolumes, which generalizes recent results in convex geometry of Kaveh-Khovanskii, Khovanskii-Timorin, Milman-Rotem and R. Schneider on reversed (or complemented) Brunn-Minkowski and Alexandrov-Fenchel inequalities. Also for toric plurisubharmonic functions in the Cegrell class, we confirm Demailly's conject…

具有在紧凸集中的指数的多项式及相关的加权极值函数 -- 近似与正则性

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Approximations and regularity

本文研究了与紧凸集相关的多项式及加权极值函数的正则性问题，通过积分、极小和极大连算子证明了相关函数的下半连续性，并推翻了Hölder连续性的已有结论。

On convex mappings

We establish a new characterization for a conformal mapping of the unit disk $\mathbb{D}$ to be convex, and identify the mappings onto a half-plane or a parallel strip as extremals. We also show that, with these exceptions, the level sets of $λ$ of the Poincaré metric $λ|dw|$ of a convex domain are strictly convex.

关于刚性 $q$-多次调和函数与 $\nathbb{C}^n$ 中的 $q$-伪凸管状域

On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$

本文证明管状域上的 $q$-多次调和函数等价于基域上的实 $q$-凸函数，并由此刻画 $q$-伪凸管状域，应用于仿射线性映射图的补集和 Reinhardt 域。

hyperbolic domains上的Thurston度量与凸包的边界

The Thurston metric on hyperbolic domains and boundaries of convex hulls

我们证明了最近点收缩是从黎曼球面上双曲域上的Thurston度量到其补的凸包边界的一致拟等距。作为推论，当区域是一致完美的时，我们得到了最近点收缩关于庞加莱度量的拟等距常数的明确界。我们还建立了Marden和Markovic的猜想，即双曲域是一致完美的当且仅当最近点收缩是Lipschitz关于Poincare度量。

指数位于紧凸集中的多项式与相关加权极值函数——Siciak-Zakharyuta定理

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Siciak-Zakharyuta theorem

本文将经典Siciak-Zakharyuta定理推广到指数限制在紧凸集$mS$中的多项式空间，证明加权极值函数等式成立当且仅当$S$的有理点稠密。

全纯水平集的双曲凸性

Hyperbolic convexity of holomorphic level sets

本文证明了全纯水平集在特定条件下关于庞加莱距离的测地凸性，并扩展了已有结果，解决了相关问题。

从有界对称域穿出的局部全纯曲线的渐近全测地性及其在代数子集单值化问题中的应用

Asymptotic total geodesy of local holomorphic curves exiting a bounded symmetric domain and applications to a uniformization problem for algebraic subsets

本文研究全纯曲线与嵌入的渐近全测地性，证明从有界对称域穿出的全纯曲线渐近全测地，推导等变全纯嵌入的全测地性，并解决域内代数子集的单值化问题，给出余紧格下全测地子集的双代数性刻画。

Poincaré关于区域双全纯不等价定理的度量版本

A metric version of Poincaré's theorem concerning biholomorphic inequivalence of domains

本文推广了Poincaré定理，证明在Kobayashi距离下，当因子数更多时，凸有界域乘积无法等距嵌入强凸域乘积，并基于度量几何建立了一般性等距嵌入障碍准则。

Invariant metrics on the Complex ellipsoid

We provide a class of geometric convex domains on which the Carathéodory-Reiffen metric, the Bergman metric, the complete Kähler-Einstein metric of negative scalar curvature are uniformly equivalent, but not proportional to each other. In a two-dimensional case, we provide a full description of curvature tensors of the Bergman metric on the weakly pseudoconvex boundary point and show that invariant metrics are proportional to each other if and only if the geometric convex domain is the Poincaré-disk.