THOMAS METTLER

We introduce a new functional ${\mathcal{E}}_{\mathfrak{p}}$ on the space of conformal structures on an oriented projective manifold $(M,\mathfrak{p})$. The non-negative 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 $(\mathrm{\Sigma },\mathfrak{p})$, we canonically construct an indefinite Kähler--Einstein structure $(h_{\mathfrak{p}},{\mathrm{\Omega }}_{\mathfrak{p}})$ on the total space $Y$ of a fibre bundle over $\mathrm{\Sigma }$ and show that a conformal structure $[g]$ is a critical point for ${\mathcal{E}}_{\mathfrak{p}}$ if and only if a certain lift $\widetilde{[g]}:(\mathrm{\Sigma },[g])\to (Y,h_{\mathfrak{p}})$ 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.

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We show that a properly convex projective structure $\mathfrak{p}$ on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if $\mathfrak{p}$ is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that $\mathfrak{p}$ admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this non-linear PDE into a transport equation, we obtain our result by applying methods from geometric inverse problems. In particular, we use an extension of a remarkable $L^2$-energy identity known as Pestov's identity to prove a vanishing theorem for the relevant transport equation.

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We introduce the notion of a minimal Lagrangian connection on the tangent bundle of a manifold and classify all such connections in the case where the manifold is a compact oriented surface of non-vanishing Euler characteristic. Combining our classification with results of Labourie and Loftin, we conclude that every properly convex projective surface arises from a unique minimal Lagrangian connection.

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