## About

I am a mathematician with research interests in projective differential geometry and its interactions with complex geometry, PDE and dynamical systems.

• Assistant professor, UniDistance Suisse
• PhD in Mathematics, Université de Fribourg (2010)
• Diploma in Physics, ETH Zürich (2005)

Email: mettler@math.ch

## Recent publications

Metrisability of projective surfaces and pseudo-holomorphic curves
Math. Z. 298 (2021)
Abstract.
We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $$\mathfrak{p}$$ and a volume form $$\sigma$$ on an oriented surface $$M$$ equip the total space of a certain disk bundle $$Z \to M$$ with a pair $$(J_{\mathfrak{p}},\mathfrak{J}_{\mathfrak{p},\sigma})$$ of almost complex structures. A conformal structure on $$M$$ corresponds to a section of $$Z \to M$$ and $$\mathfrak{p}$$ is metrisable by the metric $$g$$ if and only if $$[g] : M \to Z$$ is a pseudo-holomorphic curve with respect to $$J_{\mathfrak{p}}$$ and $$\mathfrak{J}_{\mathfrak{p},dA_g}$$.
Extremal conformal structures on projective surfaces
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XX (2020)
Abstract.
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 $$(\Sigma,\mathfrak{p})$$, we canonically construct an indefinite Kähler--Einstein structure $$(h_{\mathfrak{p}},\Omega_{\mathfrak{p}})$$ on the total space $$Y$$ of a fibre bundle over $$\Sigma$$ and show that a conformal structure $$[g]$$ is a critical point for $$\mathcal{E}_{\mathfrak{p}}$$ if and only if a certain lift $$\widetilde{[g]} : (\Sigma,[g]) \to (Y,h_{\mathfrak{p}})$$ 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.
Convex projective surfaces with compatible Weyl connection are hyperbolic (with G. Paternain)
Anal. PDE 13 (2020)
Abstract.
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.
$$\mathrm{GL}(2)$$-Structures in dimension four, $$H$$-Flatness and Integrability (with W. Krynski)
Comm. Anal. Geom. 27 (2019)
Abstract.
We show that torsion-free four-dimensional $$\mathrm{GL}(2)$$-structures are flat up to a coframe transformation with a mapping taking values in a certain subgroup $$H\subset\mathrm{SL}(4,\mathbb{R})$$, which is isomorphic to a semidirect product of the three-dimensional continuous Heisenberg group $$H_3(\mathbb{R})$$ and the Abelian group $$\mathbb{R}$$. In addition, we show that the relevant PDE system is integrable in the sense that it admits a dispersionless Lax-pair.
Minimal Lagrangian connections on compact surfaces
Adv. Math. 354 (2019)
Abstract.
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.