Positive Scalar Curvature | Gaoming Wang

Dihedral rigidity in hyperbolic 3-space

Wed, 08 Nov 2023 11:51:14 +0800

Main results

The paper proves hyperbolic analogues of Gromov’s dihedral comparison and rigidity conjectures for tetrahedra and a broad class of prisms.

Comparison

All inequalities cannot be simultaneously strict

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

Weak comparison forces the hyperbolic model

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

Every hyperbolic tetrahedron in the class is covered

All cone-type references satisfy cylinder trapping; for prisms it is equivalent to a concrete nesting condition between the two Euclidean base polygons.

Proof strategy

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.

Scalar curvature comparison of rotationally symmetric sets

Tue, 25 Apr 2023 11:51:14 +0800

Main results

This paper treats scalar-curvature rigidity for compact three-manifolds whose reference boundary is rotationally symmetric and weakly convex, including smooth, conical, and truncated models.

Euclidean rigidity

Boundary comparison determines the interior

If $R_g\geq0$, the induced boundary metric and mean curvature dominate those of the Euclidean reference, and the nonsmooth dihedral angles are no larger, then the manifold is flat under the stated pole hypotheses.

Endpoint geometries

Disks, cones, and spherical poles are all included

The theorem covers boundaries cut off by planar disks, boundaries with conical tips, and boundaries with smooth spherical poles. Standard balls and right circular cones are important special cases.

Extensions

Hyperbolic and circle-symmetric analogues

The same capillary framework yields corresponding rigidity statements in hyperbolic space and in nonpositively curved spaces with an $\mathbb S^1$ symmetry.

Proof strategy

The argument minimizes a capillary functional whose prescribed contact angle varies along the rotational boundary. Stability and Gauss–Bonnet force the minimizing disk to be infinitesimally rigid, after which a foliation propagates the equality. The main difficulty is ensuring that a nontrivial minimizer exists near a conical or spherical pole; local constant-mean-curvature barrier foliations reduce those cases to the planar-end problem.