论文标题
Loewner-Engy最小化器的确定性方法
A deterministic approach to Loewner-energy minimizers
论文作者
论文摘要
We study two minimization questions: the nature of curves $γ\subset \mathbb{H}$ which minimize the Loewner energy among all curves from 0 to a fixed $z_0 \in \mathbb{H}$, and the nature of $γ$ which minimize the Loewner energy among all curves that weld a given pair $x<0 <y$.前一个问题是由Yilin Wang部分研究的，Yilin Wang使用SLE技术来计算最小的能量并表明它是独特的。我们使用纯粹确定性的方法来重新审视这个问题，并重新启用能量公式，并获得进一步的结果，例如对驾驶功能的明确计算。我们的方法还为焊接问题以及明确的能量公式和明确的驾驶功能产生了最小化器的存在和独特性。此外，我们表明两个家庭都有“普遍性”的财产。对于焊接最小化器，这意味着有一个明确的代数曲线$γ$，因此在假想轴上的截断为$γ$或其反射$ - \overlineγ$会产生所有焊接最小化器，直到缩放。王指出她的最小化器是SLE $ _0（-8）$，但我们表明焊接最小化器是SLE $ _0（-4，-4）$。我们的结果还表明了Carto Wong的驾驶员定期定理的案例清晰度。
We study two minimization questions: the nature of curves $γ\subset \mathbb{H}$ which minimize the Loewner energy among all curves from 0 to a fixed $z_0 \in \mathbb{H}$, and the nature of $γ$ which minimize the Loewner energy among all curves that weld a given pair $x<0 <y$. The former question was partially studied by Yilin Wang, who used SLE techniques to calculate the minimal energy and show it is uniquely attained. We revisit the question using a purely deterministic methodology, and re-derive the energy formula and also obtain further results, such as an explicit computation of the driving function. Our approach also yields existence and uniqueness of minimizers for the welding question, as well as an explicit energy formula and explicit driving function. In addition, we show both families have a "universality" property; for the welding minimizers this means that there is a single, explicit algebraic curve $Γ$ such that truncations of $Γ$ or its reflection $-\overlineΓ$ in the imaginary axis generate all welding minimizers up to scaling. While Wang noted her minimizer is SLE$_0(-8)$, we show the welding minimizers are SLE$_0(-4,-4)$. Our results also show sharpness of a case of the driver-curve regularity theorem of Carto Wong.
