Decreasing Weyl's energy by connected sums with locally conformally flat manifolds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910146410577920 |
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| author | Malchiodi, Andrea Malizia, Francesco |
| author_facet | Malchiodi, Andrea Malizia, Francesco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17547 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Decreasing Weyl's energy by connected sums with locally conformally flat manifolds Malchiodi, Andrea Malizia, Francesco Differential Geometry 53C21, 53C25, 58E11, 49J99 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. |
| title | Decreasing Weyl's energy by connected sums with locally conformally flat manifolds |
| topic | Differential Geometry 53C21, 53C25, 58E11, 49J99 |
| url | https://arxiv.org/abs/2604.17547 |