The coordinate change formula for the Liouville quantum gravity metric holds for all conformal maps simultaneously

Abstract

Liouville quantum gravity (LQG) is, heuristically, a theory of random Riemannian geometry with Riemannian metric tensor $e^{γh} (\mathrm{d} x^2 + \mathrm{d} y^2)$, where $h$ is a variant of the Gaussian free field and $γ> 0$ is a parameter. If $U \subset \mathbb{C}$ is an open set, $φ\colon U \to φ(U)$ is a conformal map, and $h^φ = h \circ φ^{-1} + Q \log|(φ^{-1})'|$ (where $Q = Q(γ)$ is a parameter), then the LQG surface on $U$ defined with field $h$ is equivalent to the LQG surface on $φ(U)$ with field $h^φ$. This equivalence is meant in the sense that the area measures and distance functions on these surfaces are almost surely equivalent. It is known for the area measure that, in fact, this equivalence holds almost surely for all conformal maps $φ$ simultaneously (Sheffield-Wang 2016). We prove the corresponding result for the distance function. This makes precise the frequently used heuristic definition that a quantum surface is a random equivalence class of domains equipped with the LQG area measure and LQG distance function.

Figures (2)

• Figure 1: Illustration of Definition \ref{['defn:ImprovingTheLipschitzConstantRegularityEvents']}. Condition 1 says that if the red path is a $D_h$-geodesic, then $\mathfrak{a}_{\epsilon}^{-1} \hat{D}_{ \ref{['hphi']} }^{\epsilon} (\phi(u), \phi(v)) \leq \sup_{t \in [\tau^{-1}, \tau]} \frac{r \mathfrak{a}_{{\epsilon t}}^{-1}}{r^{\xi Q} \mathfrak{a}_{{\epsilon t}/r}^{-1}} (1 + \delta) D_h(u,v)$, and analogously with $D_h$ and $\mathfrak{a}_{\epsilon}^{-1} \hat{D}_{ \ref{['hphi']} }^{\epsilon}$ swapped. The second bullet of condition 2 says that if $D_h(u, v)$ is at least the $D_h$-distance from $u$ to the boundary of the gray annulus, then the red path cannot be a $\mathfrak{a}_{\epsilon}^{-1} \hat{D}_{ \ref{['hphi']} }^{\epsilon}$-geodesic for any $\phi \in \ref{['confmaps']}$. The first bullet of condition 2 says that if there is some $t \in [\tau^{-1}, \tau]$ such that $\mathfrak{a}_{\epsilon t}^{-1} \hat{D}_{h}^{\epsilon t} (u, v)$ is at least the $\mathfrak{a}_{\epsilon t}^{-1} \hat{D}_{h}^{\epsilon t}$-distance from $u$ to the boundary of the gray annulus, then the red path cannot be a $D_h$-geodesic. Condition 3 says there is a path (drawn in blue) around the cyan annulus with $D_h$-length at most $A$ times the $D_h$-distance across the cyan annulus, and analogously for each metric $\mathfrak{a}_{\epsilon}^{-1} \hat{D}_{ \ref{['hphi']} }^{\epsilon}$ with $\phi \in \ref{['confmaps']}$.
• Figure 2: Illustration of the proof of Proposition \ref{['prop:ImprovingTheLipschitzConstant']}. The cyan annulus is $\mathbb{A}_{\alpha r_{j-1}, r_{j-1}}(x_{j-1})$, the red path is the $D_h$-geodesic $P$, and the blue path is the path around $\mathbb{A}_{\alpha r_{j-1}, r_{j-1}}(x_{j-1})$ guaranteed by condition 3 in Definition \ref{['defn:ImprovingTheLipschitzConstantRegularityEvents']}. Time $t_j$ is the first time after $t_{j-1}$ that $P$ exits $B_{r_{j-1}}(x_{j-1})$, and $s_j$ is the last time before $t_j$ that $P$ exits $B_{\alpha r_{j-1}}(x_{j-1})$. By condition 1 in Definition \ref{['defn:ImprovingTheLipschitzConstantRegularityEvents']}, $\mathfrak{a}_{\epsilon}^{-1} \hat{D}_{ \ref{['hphi']} }^{\epsilon} (\phi(P(s_j)), \phi(P(t_j))) \leq \sup_{t \in [\tau^{-1}, \tau]} \frac{r_{j-1} \mathfrak{a}_{\epsilon t}^{-1}}{r_{j-1}^{\xi Q} \mathfrak{a}_{\epsilon t/r_{j-1}}^{-1}} (1 + \delta) (t_j - s_j)$. The blue path has $D_h$-length at most $A(t_j - s_j)$ by condition 3 in Definition \ref{['defn:ImprovingTheLipschitzConstantRegularityEvents']}. Since $P$ is a geodesic, and it crosses the blue path before $t_{j-1}$ and after $s_j$, we must have $s_j - t_{j-1} \leq A(t_j - s_j)$. This is used to show a $\frac{1}{A + 1}$-proportion of $P$ is comprised of the segments $P|_{[s_j, t_j]}$.

Theorems & Definitions (67)

• Theorem 1.1
• Lemma 2.1
• proof
• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof