Infinitely many solutions to the Yamabe problem on noncompact manifolds

TL;DR

This paper proves the existence of infinitely many solutions with constant scalar curvature on certain noncompact manifolds, expanding understanding of the Yamabe problem and its solution multiplicity.

Contribution

It demonstrates the existence of infinitely many solutions on noncompact product manifolds, including new classes of solutions and bifurcation phenomena.

Findings

Infinitely many solutions on products of closed manifolds and symmetric spaces.

Existence of infinitely many periodic solutions to the singular Yamabe problem.

Bifurcation branches associated with Bieberbach groups.

Abstract

We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, $\mathbb S^m \times\mathbb R^d$, $m\geq2$, $d\geq1$, and $\mathbb S^m\times\mathbb H^d$, $2\leq d<m$. As a consequence, we obtain infinitely many periodic solutions to the singular Yamabe problem on $\mathbb S^m\setminus\mathbb S^k$, for all $0\leq k<(m-2)/2$, the maximal range where nonuniqueness is possible. We also show that all Bieberbach groups in $Iso(\mathbb R^d)$ are periods of bifurcating branches of solutions to the Yamabe problem on $\mathbb S^m\times\mathbb R^d$, $m\geq2$, $d\geq1$.