We prove a comparison theorem for certain types of polyhedra in a 3-manifold with its scalar curvature bounded below by -6. The result confirms in some cases the Gromov dihedral rigidity conjecture in hyperbolic 3-space.
The paper proves hyperbolic analogues of Gromov’s dihedral comparison and rigidity conjectures for tetrahedra and a broad class of prisms.
Comparison
For the prescribed tetrahedra and prisms, scalar curvature greater than $-6$ and face mean curvature above the hyperbolic reference prevent every dihedral angle from being strictly smaller than its model value.
Rigidity
Under the cylinder-trapping condition, or the stated side-angle alternative, the inequalities $R_g\geq-6$, $H_g\geq H_{\mathrm{hyp}}$, and $\gamma_g\leq\gamma_{\mathrm{hyp}}$ imply that the polyhedron is isometric to one in hyperbolic space.
Geometric scope
All cone-type references satisfy cylinder trapping; for prisms it is equivalent to a concrete nesting condition between the two Euclidean base polygons.
A variational functional produces a stable capillary surface of constant mean curvature $\pm2$ with the contact angles of the reference foliation. Its second variation, together with Gauss–Bonnet, converts scalar-, mean-, and dihedral-curvature comparisons into a contradiction unless equality holds. In the equality case, an infinitesimally rigid capillary surface is propagated by a constant-mean-curvature foliation; a separate foliation construction at a vertex completes the global rigidity argument.
My research interests include Geometric Analysis and Partial Differential Equations.