## 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.