Communications in Analysis and Geometry
Volume 21 (2013)
Square-integrability of solutions of the Yamabe equation
Pages: 891 – 916
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 our paper . As an application we see that the smooth Yamabe invariant of any two connected compact seven-dimensional manifold is at least $74.5$. Similar conclusions follow in dimension $8$ and in dimensions $\geq 11$.
Published 27 December 2013