Liouville quantum gravity and the Brownian map II: Geodesics and continuity of the embedding

Abstract

We endow the 8/3-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere S2. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric dQ on a countable dense subset of S2. Here, we show that dQ a.s. extends uniquely and continuously to a metric d‾Q on all of S2. Letting d denote the Euclidean metric on S2, we show that the identity map between (S2,d) and (S2,d‾Q) is a.s. Hölder continuous in both directions. We establish several other properties of (S2,d‾Q), culminating in the fact that (as a random metric measure space) it agrees in law with the Brownian map. We establish analogous results for the Brownian disk and plane. Our proofs involve new estimates on the size and shape of QLE balls and related quantum surfaces, as well as a careful analysis of (S2,d‾Q) geodesics.