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

TL;DR

This paper proves that the transformation rule for the Liouville quantum gravity (LQG) metric under conformal maps applies simultaneously to all such maps, confirming a key heuristic in the theory of random Riemannian geometries.

Contribution

It establishes the conformal change formula for the LQG metric holds for all conformal maps at once, extending previous results known only for the area measure.

Findings

The LQG distance function transforms conformally as expected.

The result confirms the heuristic that quantum surfaces are equivalence classes under conformal maps.

The proof applies to all conformal maps simultaneously, not just a fixed one.

Abstract

Liouville quantum gravity (LQG) is, heuristically, a theory of random Riemannian geometry with Riemannian metric tensor $e^{\gamma h} (\mathrm{d} x^2 + \mathrm{d} y^2)$, where $h$ is a variant of the Gaussian free field and $\gamma > 0$ is a parameter. If $U \subset \mathbb{C}$ is an open set, $\phi \colon U \to \phi(U)$ is a conformal map, and $h^{\phi} = h \circ \phi^{-1} + Q \log|(\phi^{-1})'|$ (where $Q = Q(\gamma)$ is a parameter), then the LQG surface on $U$ defined with field $h$ is equivalent to the LQG surface on $\phi(U)$ with field $h^{\phi}$. 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 $\phi$ 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.