Veranstaltungen (Christian Lange)
Publications
Preprints

Good, but not very good orbifolds, arXiv, (2024).
Abstract. We construct examples of (effective) closed orbifolds which are covered by manifolds, but not finitely so.

Regularity of the geodesic flow on submanifolds , arXiv, (2024).
Abstract. We show that the geodesic flow and the exponential map of a C^k submanifold of R^n with k>=2 are of class C^{k1}.

(with C. Zwickler) Closed geodesics on compact orbifolds and on noncompact manifolds, arXiv, (2019).
Abstract. We study the existence of closed geodesics on compact Riemannian orbifolds, and on noncompact Riemannian manifolds in the presence of a cocompact, isometric group action. We show that every noncontractible Riemannian manifold which admits such an action, and every odddimensional, compact Riemannian orbifold has a nontrivial closed geodesic.
Refereed articles

(with L. Baracco, O. Bernardi, and M. Mazzucchelli) On the local maximizers of higher capacity ratios,
to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2024)
Abstract. We prove an analogue of the 4dimensional local Viterbo conjecture for the higher EkelandHofer capacities: on the space of 4dimensional smooth starshaped domains of unitary volume, endowed with the C^3 topology, the local maximizers of the kth EkelandHofer capacities are those domains symplectomorphic to suitable rational ellipsoids.
arXiv.

Orbifolds and manifold quotients with upper curvature bounds,
Transform. Groups (2024).
Abstract. We characterize Riemannian orbifolds with an upper curvature bound in the Alexandrov sense as reflectofolds, i.e. Riemannian orbifolds all of whose local groups are generated by reflections, with the same upper bound on the sectional curvature. Combined with a result by LytchakThorbergsson this implies that a quotient of a Riemannian manifold by a closed group of isometries has locally bounded curvature (from above) in the Alexandrov sense if and only if it is a reflectofold.
Journal version, arXiv pdf.

On continuous billiard and quasigeodesic flows characterizing alcoves and isosceles tetrahedra,
J. Lond. Math. Soc. (2) 107 (2023), no. 5, 17541779.
Abstract. We characterize fundamental domains of affine reflection groups as those polyhedral convex bodies which support a continuous billiard dynamics. We interpret this characterization in the broader context of Alexandrov geometry and prove an analogous characterization for isosceles tetrahedra in terms of continuous quasigeodesic flows. Moreover, we show an optimal regularity result for convex bodies: the billiard dynamics is continuous if the boundary is of class C^{2,1}. In particular, billiard trajectories converge to geodesics on the boundary in this case. Our proof of the latter continuity statement is based on Alexandrov geometry methods that we discuss resp. establish first.
arXiv pdf.

(with T. Soethe) Sharp systolic inequalities for rotationally symmetric 2orbifolds,
J. Fixed Point Theory Appl. 25 (2023), no. 1, Paper No. 41, 30 pp.
Abstract. We show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at socalled Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed.
Journal version, arXiv pdf.

(with M. Radeschi) How highly connected can an orbifold be?,
Rev. Mat. Iberoam. 39 (2023), no. 6, 21712186.
Abstract. On the one hand, we provide the first examples of arbitrarily highly connected (compact) bad orbifolds. On the other hand, we show that nconnected norbifolds are manifolds. The latter improves the best previously known bound of Lytchak by roughly a factor of 2. For compact orbifolds and in most dimensions we prove slightly better bounds. We obtain sharp results up to dimension 5.
Journal version, arXiv pdf.

(with C. Gorodski, A. Lytchak and R. Mendes) A diameter gap for quotients of the unit sphere,
J. Eur. Math. Soc. (JEMS) 25 (2023), no. 9, 37673793.
Abstract. We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimensionindependent positive constant.
Journal version, arXiv pdf.

(with A. Abbondandolo and M. Mazzucchelli) Higher systolic inequalities for 3dimensional contact manifolds,
J. Éc. polytech. Math. (9) 807851 (2022).
Abstract. We prove that Besse contact forms on closed connected 3manifolds, that is, contact forms with a periodic Reeb flow, are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.
Journal version, arXiv pdf.

(with T. Mettler) Deformations of the Veronese embedding and Finsler 2spheres of constant curvature,
J. Inst. Math. Jussieu 21 (2022), no. 6, 21032134.
Abstract. We establish a onetoone correspondence between Finsler structures on the 2sphere 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 CP(a1,a2)>CP(a1,(a1+a2)/2,a2) of weighted projective spaces provide examples of Finsler 2spheres of constant curvature and all geodesics closed.
Journal version, arXiv pdf.

(with A. Lytchak and C. Sämann) Lorentz meets Lipschitz,
Adv. Theor. Math. Phys. 25 (2021), no. 8, 21412170.
Abstract. We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a C^{1,1}parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an aHölder continuous Lorentzian metric admit a C^{1,a/4}parametrization.
Journal version arXiv.

(with M. Amann and M. Radeschi) Odddimensional orbifolds with all geodesics closed are covered by manifolds
Math. Ann. 380, 13551386 (2021).
Abstract. Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new examples is restricted to even dimensions. As one key ingredient we provide a characterization of orientable manifolds among orientable orbifolds in terms of characteristic classes.
Journal version, arXiv pdf.

(with H. Geiges) Erratum to "Seifert fibrations of lens spaces"  Fibrations over nonorientable bases,
Abh. Math. Semin. Univ. Hambg. 91 (2021), no. 1, 145150.
Abstract. We classify the Seifert fibrations of lens spaces where the base orbifold is nonorientable. This is an addendum to our earlier paper `Seifert fibrations of lens spaces'. We correct Lemma 4.1 of that paper and fill the gap in the classification that resulted from the erroneous lemma.
Journal version, arXiv, (2020)

(with M. Kegel) A BoothbyWang theorem for Besse contact manifolds,
Arnold Math J., 7, 225241 (2021).
Abstract. We prove a BoothbyWang type theorem for Besse Reeb flows that are not necessarily Zoll, i.e. Reeb flows all of whose orbits are periodic, but possibly with different periods. More precisely, we characterize contact manifolds whose Reeb flows are Besse as principal circleorbibundles over integral symplectic orbifolds satisfying some cohomological condition. As a corollary of this and of a result by CristofaroGardiner and Mazzucchelli we obtain a complete classification of closed Besse contact 3manifolds up to strict contactomorphism.
Journal version, arXiv pdf.

(with L. Asselle) On the rigidity of Zoll magnetic systems on surfaces,
Nonlinearity, Volume 33, Number 7, (2020).
Abstract. In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic systems such that the associated Hamiltonian flow is Zoll, i.e. every orbit is closed, on every energy level. We also prove the persistence of possibly degenerate closed geodesics under magnetic perturbations in different instances.
Journal version, arXiv pdf.

Orbifolds from a metric viewpoint,
Geom. Dedicata 209 (2020), 4357.
Abstract. We characterize Riemannian orbifolds and their coverings in terms of metric geometry. In particular, we show that the metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and that the natural projection is an orbifold covering.
Journal version, arXiv.

On metrics on 2orbifolds all of whose geodesics are closed,
J. Reine Angew. Math. 758 (2020), 6794.
Abstract. We show that the geodesic period spectrum of a Riemannian 2orbifold all of whose geodesics are closed depends, up to a constant, only on its orbifold topology and compute it. In the manifold case we recover the fact proved by Gromoll, Grove and Pries that all prime geodesics have the same length. In the appendix we partly strengthen our result in terms of conjugacy of contact forms and explain how to deduce rigidity on the real projective plane based on a systolic inequality due to Pu.
Journal version, arXiv.

When is the underlying space of an orbifold a manifold?,
Trans. Amer. Math. Soc. 372 (2019), no. 4, 27992828.
Abstract. We classify orthogonal actions of finite groups on Euclidean vector spaces for which the corresponding quotient space is a topological, homological or Lipschitz manifold, possibly with boundary. In particular, our results answer the question of when the underlying space of an orbifold is a manifold.
Journal version, arXiv.

(with U. Frauenfelder and S. Suhr) A Hamiltonian version of a result of Gromoll and Grove,
Ann. Inst. Fourier (Grenoble) 69 (2019), no. 1, 409419.
Abstract. The theorem that if all geodesics of a Riemannian twosphere are closed they are also simple closed is generalized to real Hamiltonian structures on RP^3. For reversible Finsler 2spheres all of whose geodesics are closed this implies that the lengths of all geodesics coincide.
Journal version, arXiv.

On the existence of closed geodesics on 2orbifolds,
Pacific J. Math. 294 (2), (2018), 453472.
Abstract. We show that on every compact Riemannian 2orbifold there exist infinitely many closed geodesics of positive length.
Journal version, arXiv.

(with H. Geiges) Seifert fibrations of lens spaces,
Abh. Math. Semin. Univ. Hambg. 88 (1), (2018), 122.
Abstract. We classify the Seifert fibrations of any given lens space L(p,q). We give an algorithmic construction of a Seifert fibration of L(p,q) over the base orbifold S^2(m,n) with the coprime parts of m and n arbitrarily prescribed. This algorithm produces all possible Seifert fibrations, and the equivalences between the resulting Seifert fibrations are described completely. Also, we show that all Seifert fibrations are equivalent to certain standard models.
Journal version, arXiv.

(with S. Stadler) Affine functions on Alexandrov spaces,
Math. Z. 289 (2018), no. 12, 455469.
Abstract. We show that every finitedimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.
Journal version, arXiv.

Equivariant smoothing of piecewise linear manifolds,
Math. Proc. Camb. Philos. Soc. 164 (2), (2018), 369380.
Abstract. We characterize finite groups G generated by orthogonal transformations in a finitedimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.
Journal version, arXiv.

Characterization of finite groups generated by reflections and rotations,
J. Topol. 9 (4), (2016), 11091129.
Abstract. We characterize finite groups G generated by orthogonal transformations in a finitedimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.
Journal version, arXiv.

(with M. Mikhailova), Classification of finite groups generated by reflections and rotations,
Transform. Groups 21 (4), (2016), 11551201.
Abstract. We classify finite groups generated by orthogonal transformations in a finitedimensional Euclidean space whose fixed point subspace has codimension one or two. These groups naturally arise in the study of the quotient of a Euclidean space by a finite orthogonal group and hence in the theory of orbifolds.
Journal version, arXiv
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Thesis