## Contact Information

Institut für Mathematik
Goethe-Universität Frankfurt
Robert-Mayer-Str. 10
60325 Frankfurt am Main
Germany

Office: 807
Phone: +4969-798-22596
Email:

## Aktuelle Lehre: Proseminar Analysis

Das Proseminar findet in Blockform vom 8-9. April 2020 statt. Präsenz bei allen Vorträgen ist obligatorisch.

Voraussetzung für die Teilnahme ist die Vorlesung Analysis I. Nach Auswahl eines Vortragsthemas erfolgt die Anmeldung zum Seminar per .

Besprechungstermine nach Vereinbarung

Der Vortrag inklusive Fragen soll 50-60 Minuten dauern. Für die Zuhörer/innen soll eine Zusammenfassung in Form eines Handouts von 2-4 Seiten abgegeben werden, idealerweise verfasst in Latex. Das Handout muss per E-Mail bis zum 31. März 2020 abgegeben werden.

Themenliste

Irrationale Zahlen
Das Basler Problem und die Zeta-Funktion
Die Gammafunktion
Der Fundamentalsatz der Algebra
Die Legendreschen Polynome
Mittelwerte von reellen positiven Zahlen
Konvexität und die Jensensche Ungleichung
Grundlagen der Topologie
Die Quotiententopologie
Die Cantormenge und fraktale Dimensionen
Der Banachsche Fixpunktsatz
Das Newton-Verfahren

Literaturliste
Latex Vorlage TEX Datei PDF Datei

## Upcoming Events

I co-organise the conference Geometric structures, compactifications and group actions, to be held at the Institut für Mathematik in Frankfurt, 20-24 July 2020.

## Research Interests

Projective differential geometry, complex geometry, PDE, dynamical systems.

## Preprints

• Metrisability of projective surfaces and pseudo-holomorphic curves
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 $$\mathfrak{J}_{\mathfrak{p},dA_g}$$.
• Vortices over Riemann surfaces and dominated splittings (with G. Paternain)
Abstract. 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 $$(M,g)$$.
• Geometric theory of Weyl structures (with A. Cap)
Abstract. This article studies the bundle of Weyl structures associated to a parabolic geometry. Given a parabolic geometry of any type on a smooth manifold $$M$$, this is a natural 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 $$TA$$. In the first part of the article, we prove that all elements of the theory of Weyl structures can be interpreted in terms of natural geometric operations on $$A$$. In the second part of the article, we turn our point of view around and use the relation to parabolic geometries and Weyl structures to study the intrinsic geometry on $$A$$. This geometry is rather exotic outside of the class of torsion free parabolic geometries associated to a $$|1|$$-grading (i.e. AHS structures), to which we restrict for the rest of the article. We prove that the split-signature metric provided by the almost bi-Lagrangian structure is always Einstein. 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. In the last part of the article, we show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge-Ampère equation and thus to properly convex projective structures. Analogs for other AHS structures are indicated at the end of the article.
• Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature (with C. Lange)
Abstract. 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 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 and all geodesics closed.

## Publications

18. Convex projective surfaces with compatible Weyl connection are hyperbolic (with G. Paternain)
Anal. PDE (to appear)
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.
17. Extremal conformal structures on projective surfaces
Ann. Sc. Norm. Super. Pisa Cl. Sci. (to appear)
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 the lift of $$[g]$$. 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.
16. $$\mathrm{GL}(2)$$-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.
15. Minimal Lagrangian connections on compact surfaces
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.
14. Holomorphic differentials, thermostats and Anosov flows (with G. Paternain)
Math. Ann. 373 (2019)
Abstract. 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.
13. Gauge theory on projective surfaces and anti-self-dual Einstein metrics in dimension four (with M. Dunajski)
J. Geom. Anal. 28 (2018)
Abstract. 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.
12. Characterizing classical minimal surfaces via the entropy differential (with J. Bernstein)
J. Geom. Anal. 27 (2017)
Abstract. 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.
11. Convex integration and Legendrian approximation of curves (with N. Hungerbühler, M. Wasem)
J. Convex Anal. 24 (2017)
Abstract. 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.
10. Geodesic rigidity of conformal connections on surfaces
Math. Z. 281 (2015)
Abstract. 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.
9. Two-dimensional gradient Ricci solitons revisited (with J. Bernstein)
Int. Math. Res. Not. 2015 (2015)
Abstract. 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.
8. One-dimensional projective structures, convex curves and the ovals of Benguria & Loss (with J. Bernstein)
Comm. Math. Phys. 336 (2015)
Abstract. 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.
7. Four-dimensional Kähler metrics admitting c-projective vector fields (with A. Bolsinov, V. Matveev, S. Rosemann)
J. Math. Pures Appl. 103 (2015)
Abstract. 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.
6. On Kähler metrisability of two-dimensional complex projective structures
Monatsh. Math. 174 (2014)
Abstract. 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.
5. Weyl metrisability of two-dimensional projective structures
Math. Proc. Camb. Philos. Soc. 156 (2014)
Abstract. 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.
4. Reduction of $$\beta$$-integrable $$2$$-Segre structures
Comm. Anal. Geom. 21 (2013)
Abstract. 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.
3. Local embeddability of real analytic path geometries
Differential Geom. Appl. 30 (2012)
Abstract. 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 $$\mathbb{C}^2$$.
2. Soliton solutions of the mean curvature flow and minimal hypersurfaces (with N. Hungerbühler)
Proc. Amer. Math. Soc. 140 (2012)
Abstract. Let $$(M,g)$$ be an oriented Riemannian manifold of dimension at least $$3$$ and $$X \in \mathfrak{X}(M)$$ a vector field. We show that the Monge-Ampère differential system (M.A.S.) for $$X$$-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 $$X$$ is the gradient of a function $$u$$, in which case $$\bar{g}=e^{-2u}g$$. Counterexamples to this equivalence for surfaces are also given.
1. Charges of twisted branes: the exceptional cases (with S. Fredenhagen, M. Gaberdiel)
J. High Energy Phys. 58 (2005)
Abstract. 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.