Decreasing Weyl's energy by connected sums with locally conformally flat manifolds

Abstract

We study the Weyl functional on connected sums of two four-dimensional manifolds $(M,g_M)$ and $(Z,g_Z)$, assuming $g_M$ is Bach-flat and $g_Z$ locally conformally flat. We show that if $g_M$ is neither self-dual nor anti self-dual and if $g_Z$ is of positive Yamabe class, there exists a metric $g_Y$ on $Y := M \# Z$ with Weyl energy lower than that of $g_M$ (with the trivial exception of $(Z,g_Z) = (\mathbb{S}^4, g_{\mathbb{S}^4})$). This result has a relation to a conjecture by I.Singer and has a perspective application to the minimization of Weyl's energy. The proof relies on a simultaneous interplay of $W_M^+, W_M^-$ and the topology of $Z$, and also covers some orbifold cases.