论文标题
Weil-Petersson Quasicircles的Loewner-Kufarev能量和叶子
The Loewner-Kufarev Energy and Foliations by Weil-Petersson Quasicircles
论文作者
论文摘要
我们使用loewner-kufarev方程式研究了两次穿刺的riemann sphere $ \ mathbb c \ smallsetminus \ {0 \} $的和弦 - arc jordan曲线的叶子。我们将这种叶子与沿每个叶子沿着“局部绕组”的平面上的函数相关联。我们的主要定理是，当且仅当Loewner驾驶度量$ρ$具有有限的loewner-kufarev Energy时，该功能具有有限的dirichlet能量，由$$ s（ρ）= \ frac {1} {2} {2} {2} {2} \ iint_ \ iint_ iint_ s^s^1 \ times \ times \ mathbb {r} $ nirbb {r} $ nequrefor $ρ$是$ν_t（θ）^2dθdt $的形式，否则将其设置为$ \ infty $。此外，如果这两个能量中的任何一个都是有限的，它们等于恒定因子，在这种情况下，叶叶是weil-petersson icricicircles。 能量之间的二元性有几个后果。首先是Loewner-Kufarev Energy是可逆的，也就是说，在叶面的反转和时间反转下。此外，约旦曲线的Loewner能量可以使用那些将曲线作为叶片产生曲线的度量的最小Loewner-Kufarev能量表示。这提供了Weil-Petersson准圆的新定量表征。最后，我们考虑了叶面的形状变形，并表明Loewner-Kufarev Energy满足了涉及Schwarzian衍生物的精确转化法。我们的主要定理的证明使用单位盘上的dirichlet能量空间与$ l^2（2ρ）$之间的等轴测图，我们使用Hadamard的变异公式通过Loewner-Kufarev方程表示。我们的结果与$κ$ - 参数二元性以及Schramm-loewner演变的巨大偏差以及高斯随机场相关。
We study foliations by chord-arc Jordan curves of the twice punctured Riemann sphere $\mathbb C \smallsetminus \{0\}$ using the Loewner-Kufarev equation. We associate to such a foliation a function on the plane that describes the "local winding" along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure $ρ$ has finite Loewner-Kufarev energy, defined by $$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.
