On Riemannian polyhedra with non-obtuse dihedral angles in 3-manifolds with positive scalar curvature

TL;DR

This paper classifies 3D convex polytopes that can be embedded as mean curvature convex or totally geodesic in 3-manifolds with positive scalar curvature, extending known hyperbolic results and constructing higher-dimensional examples.

Contribution

It determines the combinatorial types of realizable 3D convex polytopes with non-obtuse dihedral angles in positive scalar curvature manifolds and constructs higher-dimensional examples.

Findings

Classified all realizable 3D convex polytopes with non-obtuse dihedral angles.

Extended results analogous to Andreev's theorem to positive scalar curvature settings.

Constructed examples of such polytopes in higher dimensions.

Abstract

We determine the combinatorial types of all the 3-dimensional simple convex polytopes in R^3 that can be realized as mean curvature convex (or totally geodesic) Riemannian polyhedra with non-obtuse dihedral angles in Riemannian 3-manifolds with positive scalar curvature. This result can be considered as an analogue of Andreev's theorem on 3-dimensional hyperbolic polyhedra with non-obtuse dihedral angles. In addition, we construct many examples of such kind of simple convex polytopes in higher dimensions.