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 β1. Then it
follows from the work of Besson-Courtois-Gallot that Ο1β(S) can be
represented by a hypersurface Sβ² in N with volume less than or equal to
that of S. We study the equality case: if Ο1β(S) cannot be represented by
a hypersurface Sβ² 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.Comment: 16 pages, 2 figure