Communications in Analysis and Geometry

Volume 21 (2013)

Number 5

Square-integrability of solutions of the Yamabe equation

Pages: 891 – 916

DOI: http://dx.doi.org/10.4310/CAG.2013.v21.n5.a2

Authors

Bernd Ammann (Fakultät für Mathematik, Universität Regensburg, Germany)

Mattias Dahl (Institutionen för Matematik, Kungliga Tekniska Högskolan, Stockholm, Sweden)

Emmanuel Humbert (Laboratoire de Mathématiques et Physique Théorique, Université de Tours, France)

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 our paper [4]. 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 $\geq 11$.

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