Quantitative finiteness for maximal complex totally geodesic submanifolds of non-arithmetic ball quotients

Abstract

We prove that the number of maximal complex totally geodesic submanifolds of a smooth finite-volume non-arithmetic ball quotient is bounded linearly by its volume, with a constant depending only on the complex dimension. This gives a quantitative strengthening of the finiteness theorems of Bader--Fisher--Miller--Stover (Invent. Math., 2023) and Baldi--Ullmo (Ann. of Math., 2023).