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| Format: | Preprint |
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2015
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| Online Access: | https://arxiv.org/abs/1510.01043 |
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| _version_ | 1866929550474084352 |
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| author | Mettler, Thomas |
| author_facet | Mettler, Thomas |
| contents | We introduce a new functional $\mathcal{E}_{\mathfrak{p}}$ on the space of conformal structures on an oriented projective manifold $(M,\mathfrak{p})$. The nonnegative quantity $\mathcal{E}_{\mathfrak{p}}([g])$ measures how much $\mathfrak{p}$ deviates from being defined by a $[g]$-conformal connection. In the case of a projective surface $(Σ,\mathfrak{p})$, we canonically construct an indefinite Kähler--Einstein structure $(h_{\mathfrak{p}},Ω_{\mathfrak{p}})$ on the total space $Y$ of a fibre bundle over $Σ$ and show that a conformal structure $[g]$ is a critical point for $\mathcal{E}_{\mathfrak{p}}$ if and only if a certain lift $\widetilde{[g]} : (Σ,[g]) \to (Y,h_{\mathfrak{p}})$ is weakly conformal. In fact, in the compact case $\mathcal{E}_{\mathfrak{p}}([g])$ is -- up to a topological constant -- just the Dirichlet energy of $\widetilde{[g]}$. As an application, we prove a novel characterisation of properly convex projective structures among all flat projective structures. As a by-product, we obtain a Gauss--Bonnet type identity for oriented projective surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1510_01043 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Extremal conformal structures on projective surfaces Mettler, Thomas Differential Geometry Analysis of PDEs Geometric Topology We introduce a new functional $\mathcal{E}_{\mathfrak{p}}$ on the space of conformal structures on an oriented projective manifold $(M,\mathfrak{p})$. The nonnegative quantity $\mathcal{E}_{\mathfrak{p}}([g])$ measures how much $\mathfrak{p}$ deviates from being defined by a $[g]$-conformal connection. In the case of a projective surface $(Σ,\mathfrak{p})$, we canonically construct an indefinite Kähler--Einstein structure $(h_{\mathfrak{p}},Ω_{\mathfrak{p}})$ on the total space $Y$ of a fibre bundle over $Σ$ and show that a conformal structure $[g]$ is a critical point for $\mathcal{E}_{\mathfrak{p}}$ if and only if a certain lift $\widetilde{[g]} : (Σ,[g]) \to (Y,h_{\mathfrak{p}})$ is weakly conformal. In fact, in the compact case $\mathcal{E}_{\mathfrak{p}}([g])$ is -- up to a topological constant -- just the Dirichlet energy of $\widetilde{[g]}$. As an application, we prove a novel characterisation of properly convex projective structures among all flat projective structures. As a by-product, we obtain a Gauss--Bonnet type identity for oriented projective surfaces. |
| title | Extremal conformal structures on projective surfaces |
| topic | Differential Geometry Analysis of PDEs Geometric Topology |
| url | https://arxiv.org/abs/1510.01043 |