A polyhedron comparison theorem for 3-manifolds with positive scalar curvature

Abstract

The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is a isometric to a flat polyhedron.

Dihedral Rigidity of Parabolic Polyhedrons in Hyperbolic Spaces

“…In [22] and [23], Conjecture 1.2 was proved for a Riemannian polyhedrons that are over-Ppolyhedral, where 1) either n = 3, and P ⊂ R 3 is an arbitrary simplex;…”

“…2) each face of M is weakly strictly mean convex; 1 3) the dihedral angles between adjacent faces are all acute. Theorem 1.1 also has a rigidity statement: if n ≤ 7, and we assume all dihedral angles are not larger than π/2 in condition (3), then (M, g) is isometric to an Euclidean rectangular solid (see [22,23]). This is called the dihedral rigidity phenomenon.…”

“…Then (M, g) is isometric to a flat polyhedron in R n . Conjecture 1.2 and related problems have been studied and extended in recent years (see, e.g., [16,17,18,22,23,24,25]), leading to a range of interesting new discoveries and questions on manifolds with nonnegative scalar curvature. In this paper, we investigate the analogous polyhedral comparison principle, together with the rigidity phenomenon, for metrics with negative scalar curvature lower bound.…”

“…It also holds for more general polyhedral types, as long as the comparison model is a parabolic prism [0, 1] × P , and P ⊂ R n−1 satisfies Conjecture 1.2. By [22,23], P can be any 3-dimension simplices or n-dimensional non-obtuse prisms. See Theorem 2.3.…”

A compactness theorem for rotationally symmetric Riemannian manifolds with positive scalar curvature

“…It remains an interesting open question to prove or disprove the prism inequality on the limit space. This question is so challenging even for smooth metric spaces that it was only recently settled by Li in [Li17].…”