The Yamabe constant on noncompact manifolds

TL;DR

This paper investigates the properties of the Yamabe constant on noncompact manifolds, establishing its continuity with respect to certain topologies and exploring related concepts like the Yamabe constant at infinity.

Contribution

It proves the continuity of the Yamabe constant on noncompact manifolds and analyzes the behavior of the Yamabe constant at infinity under various topologies.

Findings

Yamabe constant depends continuously on the metric in the fine C^2-topology.

Yamabe constant at infinity is locally constant in this topology.

Continuity properties vary with different topologies on the space of metrics.

Abstract

We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe constant at infinity is even locally constant with respect to this topology. We also discuss to which extent the Yamabe constant is continuous with respect to coarser topologies on the space of Riemannian metrics.