Appears in collection : 2024 - T2 - WS2 - Group actions with hyperbolicity and measure rigidity

When can a negatively curved manifold admit infinitely many totally geodesic submanifolds of dimension at least two? I will explain some motivations for this question coming from different parts of mathematics. I will also explain a proof of the fact that a compact manifold with a real-analytic negatively curved metric admits only finitely many totally geodesic hypersurfaces, unless it is a hyperbolic manifold. And also state a more general conjecture. This is joint work with Simion Filip and Ben Lowe.