Oberseminar Geometrie, Topologie und Analysis

Oberseminar Geometrie, Topologie und Analysis

H. Geiges, A. Lytchak, G. Marinescu, S. Sabatini

Wintersemester 2017/18

Freitag, 10:30-11:30, Seminarraum 2 (Raum 204)

6.10.17 BHKM Seminar in Köln

20.10.17 Filip Misev (Marseille)
The stabilisation height of fibre surfaces

Abstract: Fibre surfaces are embedded surfaces with boundary which arise as pages of open books for the three-sphere. The simplest examples are the standard disc and the Hopf bands (embedded annuli with a full twist). Using a special operation called plumbing, fibre surfaces can be glued together to form new ones. Call a fibre surface "stable" if it can be obtained from the disc by iterated plumbing of Hopf bands. The stabilisation height measures (in terms of Hopf plumbing) how far a fibre surface is from being stable. We will show that the stabilisation height is unbounded, even among fibre surfaces of fixed genus. (Joint work with S. Baader.)

3.11.17 Joint Seminar on Complex Algebraic Geometry and Complex Analysis in Bochum

9.11.17
Thursday BHKM Seminar in Bochum

17.11.17 Ivan Izmestiev (Fribourg)
Ivory's theorem: old and new

Abstract: Ivory's lemma states that the diagonals of a curvilinear quadrilateral bounded by confocal conics (and more generally, the great diagonals of a domain bounded by confocal quadrics) have equal lengths. This lemma is used in the proof of Ivory's theorem about the potential of an equilibrium charge on an ellipsoid. In this talk, based on a joint work with Sergey Tabachnikov, we will review Blaschke's characterization of coordinate nets that satisfy Ivory's lemma and will generalize Ivory's theorem to the potential of ellipsoidal layers in the spherical and hyperbolic space.

1.12.17 Vladimir Matveev (Jena)
Binet-Legendre metric and applications of Riemannian results in Finsler geometry

Abstract: We introduce a construction that associates a Riemannian metric g_F (called the Binet-Legendre metric) to a given Finsler metric F on a smooth manifold M. The transformation F ↦ g_F is C⁰-stable and has good smoothness properties, in contrast to previously considered constructions. The Riemannian metric g_F also behaves nicely under conformal or isometric transformations of the Finsler metric F that makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named problems in Finsler geometry. In particular we extend a classical result of Wang to all dimensions. We answer a question of Matsumoto about local conformal mapping between two Berwaldian spaces and use it to investigation of essentially conformally Berwaldian manifolds. We describe all possible conformal self maps and all self similarities on a Finsler manifold, generalising the famous result of Obata to Finslerian manifolds. We also classify all compact conformally flat Finsler manifolds. We solve a conjecture of Deng and Hou on locally symmetric Finsler spaces. We prove smoothness of isometries of Holder-continuous Finsler metrics. We construct new "easy to calculate" conformal and metric invariants of Finsler manifolds.

The results are based on the papers arXiv:1104.1647, arXiv:1409.5611, arXiv:1408.6401, arXiv:1506.08935 and arXiv:1406.2924, partially joint with M. Troyanov (EPF Lausanne) and Yu. Nikolayevsky (Melbourne).

7.12.17
Thursday BHKM Seminar in Münster

8.12.17 Anna Siffert (MPI Bonn)
Some inequalities between eigenvalues of classical eigenvalue problems

Abstract: We provide several inequalities between eigenvalues of some classical eigenvalue problems on domains with C² boundary in complete Riemannian manifolds. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kuttler and Sigillito from subsets of R² to the manifold setting. This is joint work with Asma Hassannezhad.

15.12.17 Sebastian Durst
Fakultätsöffentliche Disputation

12.1.18 Martin Kerin (Münster)
Non-negative curvature on exotic spheres

Abstract: Since their discovery, there has been much interest in the question of precisely which exotic spheres admit a metric with non-negative sectional curvature. In dimension 7, Gromoll and Meyer found the first such example. It was subsequently shown by Grove and Ziller that all of the Milnor spheres admit non-negative curvature. In this talk, it will be demonstrated that the remaining exotic 7-spheres also admit non-negative curvature. This is joint work with K. Shankar and S. Goette.

19.1.18 Marco Freibert (Kiel)
On the holonomy groups obtainable by the left-invariant Hitchin and hypo flow

Abstract: The Hitchin flow in seven dimension starts with a cocalibrated G₂-structure and produces eight-dimensional Riemannian manifolds with holonomy in the exceptional holonomy group Spin(7). As the special holonomy groups Sp(2) and SU(4) are subgroups of Spin(7), these holonomies may also be obtained by the Hitchin flow. E.g. one gets holonomy in SU(4) if the initial cocalibrated G₂-structure is induced by a hypo SU(3)-structure as then the Hitchin flow is induced by the hypo flow. For both flows, we present in the left-invariant context on a Lie group some results on how properties of the initial value restrict the holonomy group of the Riemannian metric obtained by the flow. In particular, we provide properties for which one always gets holonomy equal to SU(4).

H. Geiges, 5.4.02


6.10.17	BHKM Seminar in Köln
20.10.17	Filip Misev (Marseille) The stabilisation height of fibre surfaces Abstract: Fibre surfaces are embedded surfaces with boundary which arise as pages of open books for the three-sphere. The simplest examples are the standard disc and the Hopf bands (embedded annuli with a full twist). Using a special operation called plumbing, fibre surfaces can be glued together to form new ones. Call a fibre surface "stable" if it can be obtained from the disc by iterated plumbing of Hopf bands. The stabilisation height measures (in terms of Hopf plumbing) how far a fibre surface is from being stable. We will show that the stabilisation height is unbounded, even among fibre surfaces of fixed genus. (Joint work with S. Baader.)
3.11.17	Joint Seminar on Complex Algebraic Geometry and Complex Analysis in Bochum
9.11.17 Thursday	BHKM Seminar in Bochum
17.11.17	Ivan Izmestiev (Fribourg) Ivory's theorem: old and new Abstract: Ivory's lemma states that the diagonals of a curvilinear quadrilateral bounded by confocal conics (and more generally, the great diagonals of a domain bounded by confocal quadrics) have equal lengths. This lemma is used in the proof of Ivory's theorem about the potential of an equilibrium charge on an ellipsoid. In this talk, based on a joint work with Sergey Tabachnikov, we will review Blaschke's characterization of coordinate nets that satisfy Ivory's lemma and will generalize Ivory's theorem to the potential of ellipsoidal layers in the spherical and hyperbolic space.
1.12.17	Vladimir Matveev (Jena) Binet-Legendre metric and applications of Riemannian results in Finsler geometry Abstract: We introduce a construction that associates a Riemannian metric g_F (called the Binet-Legendre metric) to a given Finsler metric F on a smooth manifold M. The transformation F ↦ g_F is C⁰-stable and has good smoothness properties, in contrast to previously considered constructions. The Riemannian metric g_F also behaves nicely under conformal or isometric transformations of the Finsler metric F that makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named problems in Finsler geometry. In particular we extend a classical result of Wang to all dimensions. We answer a question of Matsumoto about local conformal mapping between two Berwaldian spaces and use it to investigation of essentially conformally Berwaldian manifolds. We describe all possible conformal self maps and all self similarities on a Finsler manifold, generalising the famous result of Obata to Finslerian manifolds. We also classify all compact conformally flat Finsler manifolds. We solve a conjecture of Deng and Hou on locally symmetric Finsler spaces. We prove smoothness of isometries of Holder-continuous Finsler metrics. We construct new "easy to calculate" conformal and metric invariants of Finsler manifolds. The results are based on the papers arXiv:1104.1647, arXiv:1409.5611, arXiv:1408.6401, arXiv:1506.08935 and arXiv:1406.2924, partially joint with M. Troyanov (EPF Lausanne) and Yu. Nikolayevsky (Melbourne).
7.12.17 Thursday	BHKM Seminar in Münster
8.12.17	Anna Siffert (MPI Bonn) Some inequalities between eigenvalues of classical eigenvalue problems Abstract: We provide several inequalities between eigenvalues of some classical eigenvalue problems on domains with C² boundary in complete Riemannian manifolds. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kuttler and Sigillito from subsets of R² to the manifold setting. This is joint work with Asma Hassannezhad.
15.12.17	Sebastian Durst Fakultätsöffentliche Disputation
12.1.18	Martin Kerin (Münster) Non-negative curvature on exotic spheres Abstract: Since their discovery, there has been much interest in the question of precisely which exotic spheres admit a metric with non-negative sectional curvature. In dimension 7, Gromoll and Meyer found the first such example. It was subsequently shown by Grove and Ziller that all of the Milnor spheres admit non-negative curvature. In this talk, it will be demonstrated that the remaining exotic 7-spheres also admit non-negative curvature. This is joint work with K. Shankar and S. Goette.
19.1.18	Marco Freibert (Kiel) On the holonomy groups obtainable by the left-invariant Hitchin and hypo flow Abstract: The Hitchin flow in seven dimension starts with a cocalibrated G₂-structure and produces eight-dimensional Riemannian manifolds with holonomy in the exceptional holonomy group Spin(7). As the special holonomy groups Sp(2) and SU(4) are subgroups of Spin(7), these holonomies may also be obtained by the Hitchin flow. E.g. one gets holonomy in SU(4) if the initial cocalibrated G₂-structure is induced by a hypo SU(3)-structure as then the Hitchin flow is induced by the hypo flow. For both flows, we present in the left-invariant context on a Lie group some results on how properties of the initial value restrict the holonomy group of the Riemannian metric obtained by the flow. In particular, we provide properties for which one always gets holonomy equal to SU(4).