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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).
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Hu et al. (2026) studied this question.
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