A generalization of the Perelman gluing theorem and applications

TL;DR

This paper generalizes Perelman's Ricci curvature gluing theorem to intermediate curvature conditions, enabling new insights into the topology of metric spaces and conditions for metrics with positive intermediate Ricci curvature.

Contribution

It extends a key Ricci curvature gluing theorem to broader curvature conditions and explores topological and geometric applications of this generalization.

Findings

The observer moduli space of metrics with positive intermediate Ricci curvature can have non-trivial higher homotopy groups.

Provides a sufficient condition for the existence of metrics with positive intermediate Ricci curvature and totally geodesic boundary.

Abstract

We extend a positive Ricci curvature gluing theorem of Perelman to a range of positive intermediate curvature conditions, ranging from positive scalar curvature up to (and including) positive sectional curvature. As an application of this, we demonstrate that the observer moduli space of metrics with positive intermediate Ricci curvatures can have non-trivial higher homotopy groups. Further applications include deriving a sufficient condition for the existence of a metric with positive intermediate Ricci curvature and totally geodesic boundary.