We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLE$_\kappa$ loop measures for $\kappa\in(0,8)$. First, we construct rooted SLE$_\kappa$ loop measures in the Riemann sphere $\widehat{\mathbb C}$, which satisfy Mobius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parametrized by its $(1+\frac \kappa 8)$-dimensional Minkowski content. Second, by integrating rooted SLE$_\kappa$ loop measures, we construct the unrooted SLE$_\kappa$ loop measure in $\widehat{\mathbb C}$, which satisfies Mobius invariance and reversibility. Third, we extend the SLE$_\kappa$ loop measures from $\widehat{\mathbb C}$ to subdomains of $\widehat{\mathbb C}$ and to two types of Riemann surfaces using Brownian loop measures, and obtain conformal invariance or covariance of these measures. Finally, using a similar approach, we construct SLE$_\kappa$ bubble measures in simply/multiply connected domains rooted at a boundary point. The SLE$_\kappa$ loop measures for $\kappa\in(0,4]$ give examples of Malliavin-Kontsevich-Suhov loop measures for all $c\le 1$. The space-time homogeneity of rooted SLE$_\kappa$ loop measures in $\widehat{\mathbb C}$ answers a question raised by Greg Lawler.

中文翻译：

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SLE 循环措施

我们使用 SLE 的 Minkowski 内容（即自然参数化）为 $\kappa\in(0,8)$ 构造几种类型的 SLE$_\kappa$ 循环度量。首先，我们在黎曼球面 $\widehat{\mathbb C}$ 中构造有根 SLE$_\kappa$ 循环测度，当循环参数化为它的 $(1+\frac \kappa 8)$ 维 Minkowski 内容。其次，通过整合有根 SLE$_\kappa$ 循环测度，我们在 $\widehat{\mathbb C}$ 中构建无根 SLE$_\kappa$ 循环测度，满足 Mobius 不变性和可逆性。第三，我们将 SLE$_\kappa$ 循环测量从 $\widehat{\mathbb C}$ 扩展到 $\widehat{\mathbb C}$ 的子域以及使用布朗循环测量的两种黎曼曲面，并获得这些度量的共形不变性或协方差。最后，使用类似的方法，我们在以边界点为根的单/乘连接域中构建 SLE$_\kappa$ 气泡度量。$\kappa\in(0,4]$ 的 SLE$_\kappa$ 循环测度给出了所有 $c\le 1$ 的 Malliavin-Kontsevich-Suhov 循环测度的例子。有根 SLE$_ 的时空同质性$\widehat{\mathbb C}$ 中的 \kappa$ 循环测量回答了 Greg Lawler 提出的问题。