Large totally geodesic hypersurface rigidity conjecture

Let $N$ be a closed Riemannian manifold of dimension $k$, with sectional curvature bounded above by $-1$ and diameter at most $D$. A hypersurface in $N$ is totally geodesic hyperbolic if it is totally geodesic and has its hyperbolic metric of constant curvature $-1$.

Large totally geodesic hypersurface rigidity conjecture. For every $D>0$ and $k\in\mathbb{N}$, there exists $A=A(D,k)$ such that if $N$ contains a totally geodesic hyperbolic hypersurface with volume greater than $A$, then $N$ is isometric to a hyperbolic manifold.

The conjecture is motivated by the paper’s rigidity theorems, which show that sufficiently large totally geodesic hypersurfaces can force the ambient metric to be hyperbolic under additional hypotheses. The sharpness of the required assumptions remains unclear.