Linking at Infinity and Scalar Curvature Decay on Non-Compact Manifolds

TL;DR

This paper investigates how the topology at infinity influences scalar curvature decay on non-compact manifolds with positive scalar curvature, introducing new decay constraints and obstruction theories.

Contribution

It establishes that linking at infinity enforces polynomial scalar curvature decay and develops localized obstructions to positive scalar curvature at the ends.

Findings

Topological linking at infinity implies polynomial decay of scalar curvature.

Develops obstruction theory using μ-bubbles and minimal hypersurfaces.

Provides examples of metrics with quadratic scalar curvature decay.

Abstract

We study complete non-compact manifolds of positive scalar curvature, with a focus on how curvature decay is constrained by topology at infinity. Our first main result shows that topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry. This result provides a conceptual generalization of recently discovered examples of metrics with quadratic scalar curvature decay. Building on this decay mechanism, we develop an obstruction theory localized at the ends of non-compact manifolds. Using $\mu$--bubble exhaustions together with the analysis of stable minimal hypersurfaces and index theory, we obtain qualitative obstructions to uniformly positive scalar curvature on individual ends.