开型的最优 $L^2$ 延拓

关于有界平面区域上连续 $L^2$-最优 Hermite 度量的最优 $L^2$ 延拓

An optimal $L^2$ extension for continuous $L^2$-optimal Hermitian metrics

本文提出了有界平面区域上连续 $L^2$-最优 Hermitian 度量的最优 $L^2$ 延拓定理，并以此肯定回答了 Deng-Ning-Wang 和 Inayama 的两个问题。

Optimal extensions of conformal mappings from the unit disk to cardioid-type domains

The conformal mapping $f(z)=(z+1)^2 $ from $\mathbb{D}$ onto the standard cardioid has a homeomorphic extension of finite distortion to entire $\mathbb{R}^2 .$ We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from $\mathbb{D}$ onto cardioid-type domains.

关于全纯向量丛最优 $L^2$ 延拓定理的一个注记

A Remark on Optimal $L^2$ extension theorem for holomorphic vector bundles

本文在弱拟凸 Kähler 流形上建立了带光滑 Hermite 度量的全纯向量丛在连续增益条件下的最优 $L^2$ 延拓定理，统一了 Guan-Zhou 与 Zhou-Zhu 的已有结果。

Optimal extension of Lipschitz embeddings in the plane

We prove that every bi-Lipschitz embedding of the unit circle into the plane can be extended to a bi-Lipschitz map of the unit disk with linear bounds on the constants involved. This answers a question raised by Daneri and Pratelli. Furthermore, every Lipschitz embedding of the circle extends to a Lipschitz homeomorphism of the plane, again with a linear bound on the constant.

乘子子模层与Lempert的一个问题

Multiplier Submodule Sheaves and a problem of Lempert

该研究建立了全纯向量丛上Nakano半正奇异Hermite度量的$L^2$延拓定理及相关乘子子模层的性质，基于Deng-Ning-Wang-Zhou的Nakano正性刻画，正面解决了Lempert关于递增度量极限下Nakano半正性保持的问题。

Optimal $L^2$-extensions on tube domains and a simple proof of Prékopa's theorem

We prove the optimal $L^2$-extension theorem of Ohsawa-Takegoshi type on a tube domain. As an application, we give a simple proof of Prékopa's theorem.

Optimal $L^2$ extension for holomorphic vector bundles with singular hermitian metrics

In the present paper, we study the properties of singular Nakano positivity of singular hermitian metrics on holomorphic vector bundles, and establish an optimal $L^2$ extension theorem for holomorphic vector bundles with singular hermitian metrics on weakly pseudoconvex Kähler manifolds. As applications, we give a necessary condition for the holding of the equality in optimal $L^2$ extension theorem, and present singular hermitian holomorphic vector bundle versions of some $L^2$ extension theorems with optimal estimate.

Griffiths正性、多重调和性与平坦性的刻画

Characterizations of Griffiths Positivity, Pluriharmonicity and Flatness

该研究推广了Deng-Ning-Wang-Zhou的相关结论，通过$L^2$-延拓条件定量刻画Griffiths正性，证明$\u211d$-值可测函数的多重调和性、带奇性埃尔米特度量的平坦性均等价于最优$L^p$-延拓条件的等式部分，正面解答了Inayama的猜想。

A solution of an $L^{2}$ extension problem with optimal estimate and applications

In this paper, we prove an $L^2$ extension theorem with optimal estimate in a precise way, which implies optimal estimate versions of various well-known $L^2$ extension theorems. As applications, we give proofs of a conjecture of Suita on the equality condition in Suita's conjecture, the so-called $L-$conjecture, and the extended Suita conjecture. As other applications, we give affirmative answer to a question by Ohsawa about limiting case for the extension operators between the weighted Bergman spaces, and we present a relation of our result to Berndtsson's important result on log-plurisubhar…

关于通过Ohsawa--Takegoshi延拓定理刻画多重调和函数的一个注记

A note on characterizing pluriharmonic functions via the Ohsawa--Takegoshi extension theorem

本文证明连续函数为多重调和函数当且仅当其作为权重时最优Ohsawa--Takegoshi $L^2$-延拓定理的等式部分成立，部分解决了作者提出的猜想。

Optimal $L^2$ extension of sections from subvarieties in weakly pseudoconvex manifolds

In this paper, we obtain optimal $L^2$ extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex Kähler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular.

Optimal constant in an L^2 extension problem and a proof of a conjecture of Ohsawa

In this paper, we solve the optimal constant problem in the setting of Ohsawa's generalized $L^{2}$ extension theorem. As applications, we prove a conjecture of Ohsawa and the extended Suita conjecture, we also establish some relations between Bergman kernel and logarithmic capacity on compact and open Riemann surfaces.

Complex embeddings, Toeplitz operators and transitivity of optimal holomorphic extensions

In a setting of a complex manifold with a fixed positive line bundle and a submanifold, we consider the optimal Ohsawa-Takegoshi extension operator, sending a holomorphic section of the line bundle on the submanifold to the holomorphic extension of it on the ambient manifold with the minimal $L^2$-norm. We show that for a tower of submanifolds in the semiclassical setting, i.e. when we consider a large tensor power of the line bundle, the extension operators satisfy transitivity property modulo some small defect, which can be expressed through Toeplitz type operators. We calculate the first si…

全纯射流沿子流形的最优全纯延拓的渐近性

The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds

本文研究沿子流形的全纯射流的$L^2$-最优延拓算子的渐近公式，通过分析Schwartz核与Bergman投影，证明了射流空间上两个自然范数的渐近等距。

Positivity of holomorphic vector bundles in terms of $L^p$-conditions of $\bar\partial$

We study the positivity properties of Hermitian (or even Finsler) holomorphic vector bundles in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic objects. To this end, we introduce four conditions, called the optimal $L^p$-estimate condition, the multiple coarse $L^p$-estimate condition, the optimal $L^p$-extension condition, and the multiple coarse $L^p$-extension condition, for a Hermitian (or Finsler) vector bundle $(E,h)$. The main result of the present paper is to give a characterization of the Nakano positivity of $(E,h)$ via the optimal $L^2$-estimate condit…

Extension of holomorphic functions defined on non reduced analytic subvarieties

The goal of this contribution is to investigate L${}^2$ extension properties for holomorphic sections of vector bundles satisfying weak semi-positivity properties. Using techniques borrowed from recent proofs of the Ohsawa-Takegoshi extension theorem, we obtain several new surjectivity results for the restriction morphism to a non necessarily reduced subvariety, provided the latter is defined as the zero variety of a multiplier ideal sheaf. These extension results come with precise L${}^2$ estimates and (probably) optimal curvature conditions.