THOMAS METTLER

Given a prescription of unparametrised paths on a manifold \(M\), one path for each tangent direction, we may ask whether these paths agree with the geodesics of a Riemannian metric on \(M\). Generically, this is not the case. Motivated by this fact, we introduce a non-negative functional \(\mathcal{E}\) on the space of Riemannian metrics on \(M\) so that \(\mathcal{E}(g)=0\) if and only if the geodesics of the metric \(g\) agree with the prescribed paths. We compute the variational equations for \(\mathcal{E}\) and show that the conformal variational equation is, perhaps surprisingly, of Yamabe type. This allows us to obtain existence results for conformally critical points of \(\mathcal{E}\). In particular, in the surface case, every conformal class contains a conformally critical metric, unique up to homothety. As a by-product, we establish that the Blaschke metric of a properly convex projective surface is a critical point for \(\mathcal{E}\).

This note summarizes results obtained while the author was a member of the SPP 2026 “Geometry at Infinity” by DFG, focusing on the interplay between projective differential geometry and twistor theory on surfaces. Specifically, we provide a new twistorial perspective on the classical metrizability problem for projective structures, showing that a projective surface admits a compatible Riemannian metric if and only if the associated conformal structure embeds as both a holomorphic curve into its twistor space as well as a pseudo-holomorphic curve into the twistor space with a suitably modified almost complex structure. We also establish that while every projective surface locally admits a compatible Weyl connection, no such connection exists globally on non-hyperbolic properly convex projective surfaces — in particular, such surfaces are not metrizable. Additionally, we explore the duality between Finsler \(2\)-spheres of constant curvature with all geodesics closed and positive Besse–Weyl structures on weighted projective lines. This correspondence, realized through the notion of the dual of a path geometry, provides new examples of such Finsler metrics via deformations of the Veronese embedding.

We introduce the notion of a flat extension of a connection \(\theta\) on a principal bundle. Roughly speaking, \(\theta\) admits a flat extension if it arises as the pull-back of a component of a Maurer–Cartan form. For trivial bundles over closed oriented \(3\)-manifolds, we relate the existence of certain flat extensions to the vanishing of the Chern–Simons invariant associated with \(\theta\). As an application, we recover the obstruction of Chern–Simons for the existence of a conformal immersion of a Riemannian \(3\)-manifold into Euclidean \(4\)-space. In addition, we obtain corresponding statements for a Lorentzian \(3\)-manifold, as well as a global obstruction for the existence of an equiaffine immersion into \(\mathbb{R}^4\) of a \(3\)-manifold that is equipped with a torsion-free connection preserving a volume form.

Using the Schwarzian derivative we construct a sequence \(\left(P_{\ell}\right)_{\ell \geqslant 2}\) of meromorphic differentials on every non-flat oriented minimal surface in Euclidean \(3\)-space. The differentials \(\left(P_{\ell}\right)_{\ell \geqslant 2}\) are invariant under all deformations of the surface arising via the Weierstrass representation and depend on the induced metric and its derivatives only. A minimal surface is said to have degree \(n\) if its \(n\)-th differential is a polynomial expression in the differentials of lower degree. We observe that several well-known minimal surfaces have small degree, including Enneper’s surface, the helicoid/catenoid and the Scherk – as well as the Schwarz family. Furthermore, it is shown that locally and away from umbilic points every minimal surface can be approximated by a sequence of minimal surfaces of increasing degree.

In earlier work we have shown that for certain geometric structures on a smooth manifold \(M\) of dimension \(n\), one obtains an almost para-Kähler–Einstein metric on a manifold \(A\) of dimension \(2n\) associated to the structure on \(M\). The geometry also associates a diffeomorphism between \(A\) and \(T^*M\) to any torsion-free connection compatible with the geometric structure. Hence we can use this construction to associate to each compatible connection an almost para-Kähler–Einstein metric on \(T^*M\). In this short article, we discuss the relation of these metrics to Patterson–Walker metrics and derive explicit formulae for them in the cases of projective, conformal and Grassmannian structures.

Given a parabolic geometry on a smooth manifold \(M\), we study a natural affine bundle \(A \to M\), whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on \(A\), which induces an almost bi-Lagrangian structure on \(A\) and a compatible linear connection on \(TA\). We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with non-zero scalar curvature, provided the parabolic geometry is torsion-free and \(|1|\)-graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in \(A\). For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. We also show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge-Ampere equation and thus to properly convex projective structures.

We establish a one-to-one correspondence between Finsler structures on the \(2\)-sphere with constant curvature \(1\) and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics are closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding \(\mathbb{CP}(a_1,a_2)\to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)\) of weighted projective spaces provide examples of Finsler \(2\)-spheres of constant curvature whose geodesics are all closed.

We associate a flow \(\phi\) to a solution of the vortex equations on a closed oriented Riemannian 2-manifold \((M,g)\) of negative Euler characteristic and investigate its properties. We show that \(\phi\) always admits a dominated splitting and identify special cases in which \(\phi\) is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of \((M,g)\).

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 \(\mathfrak{J}_{\mathfrak{p},dA_g}.\)

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.

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.

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.

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.

We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian \(2\)-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties. In particular, we prove that they admit a dominated splitting and we identify special cases in which the flows are Anosov. In the latter case, we study when they admit an invariant measure in the Lebesgue class and the regularity of the weak foliations.

Given a projective structure on a surface \(N\), we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space \(M\) of a certain rank \(2\) affine bundle \(M \to N\). The Einstein metric has anti-self-dual conformal curvature and admits a parallel field of anti-self-dual planes. We show that locally every such metric arises from our construction unless it is conformally flat. The homogeneous Einstein metric corresponding to the flat projective structure on \(\mathbb{RP}^2\) is the non-compact real form of the Fubini-Study metric on \(M=\mathrm{SL}(3, \mathbb{R})/\mathrm{GL}(2, \mathbb{R})\). We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.

We introduce on any smooth oriented minimal surface in Euclidean \(3\)-space a meromorphic quadratic differential, \(P\), which we call the entropy differential. This differential arises naturally in a number of different contexts. Of particular interest is the realization of its real part as a conservation law for a natural geometric functional – which is, essentially, the entropy of the Gauss curvature. We characterize several classical surfaces – including Enneper’s surface, the catenoid and the helicoid – in terms of \(P\). As an application, we prove a novel curvature estimate for embedded minimal surfaces with small entropy differential and an associated compactness theorem.

Using convex integration we give a constructive proof of the well-known fact that every continuous curve in a contact \(3\)-manifold can be approximated by a Legendrian curve.

We show that a conformal connection on a closed oriented surface \(\Sigma\) of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on \(\Sigma\) determine the metric up to constant rescaling. It is also shown that every conformal connection on the \(2\)-sphere lies in a complex \(5\)-manifold of conformal connections, all of which share the same unparametrised geodesics.

In this note, we complete the classification of the geometry of non-compact two-dimensional gradient Ricci solitons. As a consequence, we obtain two corollaries: First, a complete two-dimensional gradient Ricci soliton has bounded curvature. Second, we give examples of complete two-dimensional expanding Ricci solitons with negative curvature that are topologically disks and are not hyperbolic space.

Benguria and Loss have conjectured that, amongst all smooth closed curves in \(\mathbb{R}^2\) of length \(2\pi\), the lowest possible eigenvalue of the operator \(L=-\Delta+\kappa^2\) is \(1\). They observed that this value was achieved on a two-parameter family, \(\mathcal{O}\), of geometrically distinct ovals containing the round circle and collapsing to a multiplicity-two line segment. We characterize the curves in \(\mathcal{O}\) as absolute minima of two related geometric functionals. We also discuss a connection with projective differential geometry and use it to explain the natural symmetries of all three problems.

A vector field on a Kähler manifold is called c-projective if its flow preserves the \(J\)-planar curves. We give a complete local classification of Kähler real 4-dimensional manifolds that admit an essential c-projective vector field. An important technical step is a local description of 4-dimensional c-projectively equivalent metrics of arbitrary signature. As an application of our results, we prove the natural analog of the classical Yano-Obata conjecture in the pseudo-Riemannian 4-dimensional case.

We derive necessary conditions for a complex projective structure on a complex surface to arise via the Levi-Civita connection of a (pseudo-)Kähler metric. Furthermore we show that the (pseudo-)Kähler metrics defined on some domain in the projective plane which are compatible with the standard complex projective structure are in one-to-one correspondence with the hermitian forms on \(\mathbb{C}^3\) whose rank is at least two. This is achieved by prolonging the relevant finite-type first order linear differential system to closed form. Along the way we derive the complex projective Weyl and Liouville curvature using the language of Cartan geometries.

We show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the ‘twistor’ bundle of conformal inner products having holomorphic image. The second solution allows to use standard results in algebraic geometry to show that the Weyl connections on the two-sphere whose geodesics are the great circles are in one-to-one correspondence with the smooth quadrics without real points in the complex projective plane.

We show that locally every \(\beta\)-integrable \((2,n)\)-Segre structure can be reduced to a torsion-free \(S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\)-structure. This is done by observing that such reductions correspond to sections with holomorphic image of a certain ‘twistor bundle’. For the homogeneous \((2,n)\)-Segre structure on the oriented \(2\)-plane Grassmannian, the reductions are shown to be in one-to-one correspondence with the smooth quadrics \(Q \subset \mathbb{CP}^{n+1}\) without real points.

An almost complex structure \(\mathfrak{J}\) on a \(4\)-manifold \(X\) may be described in terms of a rank \(2\) vector bundle \(\Lambda_{\mathfrak{J}} \subset \Lambda^2TX^*.\) We call a pair of line subbundles \(L_1,L_2\) of \(\Lambda^2TX^*\) a splitting of \(\mathfrak{J}\) if \(\Lambda_{\mathfrak{J}}=L_1\oplus L_2.\) A hypersurface \(M \subset X\) satisfying a nondegeneracy condition inherits a CR-structure from \(\mathfrak{J}\) and a path geometry from the splitting \((L_1,L_2).\) Using the Cartan-Kähler theorem we show that locally every real analytic path geometry is induced by an embedding into \(\mathbb{C}^2\) equipped with the splitting generated by the real and imaginary part of \(\mathrm{d} z^1\wedge \mathrm{d} z^2.\) As a corollary we obtain the well-known fact that every \(3\)-dimensional nondegenerate real analytic CR-structure is locally induced by an embedding into \(\mathbb{C}^2.\)

Let \((M,g)\) be an oriented Riemannian manifold of dimension at least \(3\) and \(\mathbf{X} \in \mathfrak{X}(M)\) a vector field. We show that the Monge-Ampère differential system (M.A.S.) for \(\mathbf{X}\)-pseudosoliton hypersurfaces on \((M,g)\) is equivalent to the minimal hypersurface M.A.S. on \((M,\bar{g})\) for some Riemannian metric \(\bar{g}\), if and only if \(\mathbf{X}\) is the gradient of a function \(u\), in which case \(\bar{g}=e^{-2u}g\). Counterexamples to this equivalence for surfaces are also given.

The charges of the twisted D-branes for the two exceptional cases (\(\mathrm{SO}(8)\) with the triality automorphism and \(E_6\) with charge conjugation) are determined. To this end the corresponding NIM-reps are expressed in terms of the fusion rules of the invariant subalgebras. As expected the charge groups are found to agree with those characterising the untwisted branes.