弦Loewner链的拟共形形变
Quasiconformal deformations of chordal Loewner chains
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中文总结 AI 辅助
本文推导弦Loewner链拟共形形变下驱动函数与半平面容量的一阶变分公式，结合Stoilow分解完成证明，并通过映射伸缩关系导出渐近归一化下的对应公式。
中文摘要 AI 辅助
我们建立了弦Loewner链在拟共形形变下驱动函数与半平面容量的一阶变分公式。在固定0、1和∞的标准归一化下，该公式对所有有界Beltrami微分成立。若曲线具有正面积，公式包含沿曲线的额外积分项；当曲线面积为零或微分在曲线上取零值时，这些项消失。证明采用归一化Ahlfors-Bers变分结合适配于狭缝域的Stoilow分解。随后，我们考虑对称延拓至复平面的微分的渐近归一化，其形式为ν(z)=a(\bar{z}/z)^{n-1}+ν₀(z)，其中a∈(-1,1)，n≥1为整数，ν₀∈L^r(\mathbb{C})且0<r<2。渐近归一化映射与标准归一化映射相差一个正伸缩，利用该关系，第二种归一化下的变分公式可直接由标准归一化情形导出。
英文摘要
We establish first-order variational formulas for the driving function and the half-plane capacity of a chordal Loewner chain under quasiconformal deformations. Under the standard normalization fixing $0,1$ and $\infty$, the formulas hold for every bounded Beltrami differential. If the curve has positive area, the formulas contain additional integrals over the curve; these terms vanish when the curve has zero area or when the differential is extended by zero on the curve. The proof uses the normalized Ahlfors--Bers variation together with a Stoilow factorization adapted to the slit domain. We then consider the asymptotic normalization for differentials whose symmetric extensions to $\mathbb{C}$ have the form $ν(z)=a(\bar z/z)^{n-1}+ν_0(z)$, where $a\in(-1,1)$, $n\ge1$ is an integer, and $ν_0\in L^r(\mathbb{C})$ for $0<r<2$. The asymptotically normalized map differs from the standardly normalized map by a positive dilation. Using this relation, the variational formulas in the second normalization follow directly from the standardly normalized case.
发表机构
- Shenzhen University（深圳大学）
- Chongqing University（重庆大学）
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