Gluing Riemannian manifolds with curvature operators at least k

TL;DR

This paper presents a method for gluing Riemannian manifolds with curvature bounds, ensuring the resulting manifold maintains a curvature operator at least k, under certain boundary conditions.

Contribution

It introduces a technique to preserve curvature bounds when gluing manifolds with curvature operators at least k, extending to various curvature notions.

Findings

Curvature operator bounds are preserved under gluing with positive semidefinite second fundamental forms.

Results extend to Ricci, scalar, bi, isotropic, and flag curvatures.

The method allows for small error adjustments in curvature bounds after gluing.

Abstract

We glue two manifolds which have curvature operators at least k (in the sense of eigenvalues) along their common boundary. We show that if the sum of the second fundamental forms of the boundary is positive semidefinite, then the curvature operator of the resulting manifold is at least k up to an arbitrarily small error term. Similar results hold for Ricci, scalar, bi, isotropic and flag curvature, respectively.