Loewner-Kufarev 能量与 Weil-Petersson 拟圆周的叶状结构
The Loewner-Kufarev Energy and Foliations by Weil-Petersson Quasicircles
AI总结：
本文使用 Loewner-Kufarev 方程研究黎曼球面的弦弧 Jordan 曲线叶状结构，证明了局部卷绕函数的 Dirichlet 能量与 Loewner 驱动测度的 Loewner-Kufarev 能量等价，并据此给出了 Weil-Petersson 拟圆周的新定量刻画。
摘要：
我们使用 Loewner-Kufarev 方程研究了两次穿孔黎曼球面 $\mathbb C \smallsetminus \{0\}$ 的弦弧 Jordan 曲线叶状结构。我们将这样一个函数关联到该叶状结构上，该函数描述了沿每条叶的“局部卷绕”。我们的主要定理是，该函数具有有限的 Dirichlet 能量当且仅当 Loewner 驱动测度 $ρ$ 具有有限的 Loewner-Kufarev 能量，其定义为 $$S(ρ) = \frac{1}{2}\iint_{S^1 \times \mathbb{R}} ν_t'(θ)^2 \\, d θd t$$ 只要 $ρ$ 具有形式 $ν_t(θ)^2 d θd t$，否则设为 $\infty$。此外，如果这两种能量中的任何一种是有限的，它们相等至多相差一个常数因子，在这种情况下，叶状结构的叶是 Weil-Petersson 拟圆周。这种能量之间的对偶性有几个推论。首先，Loewner-Kufarev 能量是可逆的，即在叶状结构的反演和时间反转下不变。此外，Jordan 曲线的 Loewner 能量可以使用那些生成该曲线作为叶的测度的最小 Loewner-Kufarev 能量来表达。这提供了 Weil-Petersson 拟圆周的一个新的定量刻画。最后，我们考虑叶状结构的共形畸变，并表明 Loewner-Kufarev 能量满足一个涉及 Schwarzian 导数的精确变换律。我们主要定理的证明使用了单位圆盘上的 Dirichlet 能量空间与 $L^2(2ρ)$ 之间的等距同构，该同构是我们使用通过 Loewner-Kufarev 方程表达的 Hadamard 变分公式构造的。我们的结果与 $κ$-参数对偶性以及与高斯随机场耦合的 Schramm-Loewner 演化的的大偏差相关。
英文摘要：
We study foliations by chord-arc Jordan curves of the twice punctured Riemann sphere $\mathbb C \smallsetminus \{0\}$ using the Loewner-Kufarev equation. We associate to such a foliation a function on the plane that describes the "local winding" along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure $ρ$ has finite Loewner-Kufarev energy, defined by $$S(ρ) = \frac{1}{2}\iint_{S^1 \times \mathbb{R}} ν_t'(θ)^2 \, d θd t$$ whenever $ρ$ is of the form $ν_t(θ)^2 d θd t$, and set to $\infty$ otherwise. Moreover, if either of these two energies is finite they are equal up to a constant factor, and in this case, the foliation leaves are Weil-Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner-Kufarev energy is reversible, that is, invariant under inversion and time-reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner-Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil-Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner-Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and $L^2(2ρ)$ that we construct using Hadamard's variational formula expressed by means of the Loewner-Kufarev equation. Our results are related to $κ$-parameter duality and large deviations of Schramm-Loewner evolutions coupled with Gaussian random fields.
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