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

We endow the $\sqrt{8/3}$-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 ${\mathbf S}^2$. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric $d_{\mathcal Q}$ on a countable dense subset of ${\mathbf S}^2$. Here we show that $d_{\mathcal Q}$ a.s. extends uniquely and continuously to a metric $\bar{d}_{\mathcal Q}$ on all of ${\mathbf S}^2$. Letting $d$ denote the Euclidean metric on ${\mathbf S}^2$, we show that the identity map between $({\mathbf S}^2, d)$ and $({\mathbf S}^2, \bar{d}_{\mathcal Q})$ is a.s. H\"older continuous in both directions. We establish several other properties of $({\mathbf S}^2, \bar{d}_{\mathcal Q})$, 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 $({\mathbf S}^2, \bar{d}_{\mathcal Q})$ geodesics.