2023/06/02 by Ben Lowe, Lowe, Ben
Mathematics · #53A10 #53C24 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2306.01254
openalex publication_date 2023/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a closed hyperbolic manifold containing a totally geodesic hypersurface S, and let N be a closed Riemannian manifold homotopy equivalent to M with sectional curvature bounded above by -1. Then it follows from the work of Besson-Courtois-Gallot that π1(S) can be represented by a hypersurface S' in N with volume less than or equal to that of S. We study the equality case: if π1(S) cannot be represented by a hypersurface S' in N with volume strictly smaller than that of S, then must N be isometric to M? We show that many such S are rigid in the sense that the answer to this question is positive. On the other hand, we construct examples of S for which the answer is negative.