Dihedral rigidity for submanifolds of warped product manifolds

TL;DR

This paper establishes dihedral extremality and rigidity theorems for certain submanifolds with polyhedral boundaries in warped product manifolds, broadening understanding of geometric rigidity beyond classical assumptions.

Contribution

It introduces new dihedral rigidity results applicable to submanifolds with complex face orientations in warped product spaces, including cases with flat leaves and non-log-concave warping functions.

Findings

Proves dihedral extremality for submanifolds with polyhedral boundary

Extends rigidity results to non-orthogonal face configurations

Includes a special case for hyperbolic polyhedra with flat leaves

Abstract

In this paper, we prove a dihedral extremality and rigidity theorem for a large class of codimension zero submanifolds with polyhedral boundary in warped product manifolds. We remark that the spaces considered in this paper are not necessarily warped product manifolds themselves. In particular, the results of this paper are applicable to submanifolds (of warped product manifolds) with faces that are neither orthogonal nor parallel to the radial direction of the warped product metric. Generally speaking, the dihedral rigidity results require the leaf of the underlying warped space to have positive Ricci curvature and the warping function to be strictly log-concave. Nevertheless, we prove a dihedral rigidity theorem for a large class of hyperbolic polyhedra, where the leaf of the underlying warped product space is flat and the warping function is not strictly log-concave.