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Thomas Mettler

Publications and source records attributed to Thomas Mettler.

At least 19 recordsLinked to original sources
arXiv2026

Prescribing geodesics and a variational problem for Riemannian metrics

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 $\mathcal{E}$.

math.DG↗
arXiv2025

Flat extensions of principal connections and the Chern-Simons $3$-form

We introduce the notion of a flat extension of a connection $θ$ on a principal bundle. Roughly speaking, $θ$ 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 $θ$. 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.

math.DG↗
arXiv2024

The Schwarzian derivative and the degree of a classical minimal surface

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.

math.DG↗
arXiv2023

Induced almost para-K\"ahler Einstein metrics on cotangent bundles

In earlier work we have shown that for certain geometric structures on a smooth manifold $M$ of dimension $n$, one obtains an almost para-K\"ahler--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-K\"ahler--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.

math.DG↗
arXiv2022

Geometric Theory of Weyl 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.

math.DG↗
arXiv2021

Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature

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.

math.DG↗
arXiv2020

Vortices over Riemann surfaces and dominated splittings

We associate a flow $ϕ$ 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 $ϕ$ always admits a dominated splitting and identify special cases in which $ϕ$ 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 $(M,g)$.

math.DG↗
arXiv2020

Metrisability of projective surfaces and pseudo-holomorphic curves

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 $σ$ on an oriented surface $M$ equip the total space of a certain disk bundle $Z\to M$ with a pair $(J_{\mathfrak{p}},\mathfrak{J}_{\mathfrak{p},σ})$ 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 $\mathfrak{J}_{\mathfrak{p},dA_g}$.

math.DG↗
arXiv2019

Minimal Lagrangian connections on compact surfaces

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.

math.DG↗
arXiv2019

Convex projective surfaces with compatible Weyl connection are hyperbolic

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.

math.DG↗
arXiv2019

First Operation of a Resistive Shell Liquid Argon Time Projection Chamber -- A new Approach to Electric-Field Shaping

We present a new technology for the shaping of the electric field in Time Projection Chambers (TPCs) using a carbon-loaded polyimide foil. This technology allows for the minimisation of passive material near the active volume of the TPC and thus is capable to reduce background events originating from radioactive decays or scattering on the material itself. Furthermore, the high and continuous electric resistivity of the foil limits the power dissipation per unit area and minimizes the risks of damages in the case of an electric field breakdown. Replacing the conventional field cage with a resistive plastic film structure called 'shell' decreases the number of components within the TPC and therefore reduces the potential points of failure when operating the detector. A prototype liquid argon (LAr) TPC with such a resistive shell and with a cathode made of the same material was successfully tested for long term operation with electric field values up to about 1.5 kV/cm. The experiment shows that it is feasible to successfully produce and shape the electric field in liquefied noble-gas detectors with this new technology.

physics.ins-det↗
arXiv2018

Extremal conformal structures on projective surfaces

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.

math.DG↗
arXiv2018

Holomorphic differentials, thermostats and Anosov flows

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.

math.DG↗
arXiv2017

Gauge theory on projective surfaces and anti-self-dual Einstein metrics in dimension four

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 $M=\mathrm{SL}(3, \mathrm{R})/\mathrm{GL}(2, \mathrm{R})$. We also show how our construction relates to a certain gauge-theoretic equation introduced by Calderbank.

math.DG↗
arXiv2016

$\mathrm{GL}(2)$-structures in dimension four, $H$-flatness and integrability

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.

math.DG↗
arXiv2016

Characterizing classical minimal surfaces via the entropy differential

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.

math.DG↗
arXiv2015

Geodesic rigidity of conformal connections on surfaces

We show that a conformal connection on a closed oriented surface $Σ$ 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 $Σ$ 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.

math.DG↗