Andres Contreras Hip, University of Chicago
Liouville quantum gravity is a canonical model for random surfaces conjectured to be the scaling limit of various planar maps. Since curvature is a central concept in Riemannian geometry, it is natural to ask whether this can be extended to LQG surfaces. In this talk, we introduce a notion of Gaussian curvature for LQG surfaces, despite their low regularity, and study the relations with its discrete counterparts. We conjecture that this definition of Gaussian curvature is the scaling limit of the discrete curvature. In support of this conjecture, we prove that the discrete curvature on the $\epsilon$-CRT map with a Poisson vertex set integrated with a smooth test function is of order $\epsilon^{o(1)}$, and show the convergence of the total discrete curvature on a CRT map cell when scaled by $\epsilon^{1/4}$. Joint work with E. Gwynne.