Square-integrability of solutions of the Yamabe equation

TL;DR

This paper proves that solutions to the Yamabe equation on certain non-compact manifolds that are bounded and in a specific L^p space are also in L^2, with implications for the Yamabe invariant in various dimensions.

Contribution

It establishes L^p-L^2 integrability for Yamabe solutions on non-compact manifolds, providing explicit constants for the Yamabe invariant's surgery-monotonicity formula.

Findings

Solutions are L^2 if bounded and in L^{2n/(n-2)}.

Yamabe invariant of 2-connected 7-manifolds is at least 74.5.

Similar bounds hold in dimensions 8 and ≥11.

Abstract

We show that solutions of the Yamabe equation on certain n-dimensional non-compact Riemannian manifolds which are bounded and L^p for p=2n/(n-2) are also L^2. This L^p-L^2-implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in a previous article of the authors. As an application we see that the smooth Yamabe invariant of any 2-connected compact 7-dimensional manifold is at least 74.5. Similar conclusions follow in dimension 8 and in dimensions at least 11.