Oberseminar Geometrie, Topologie und Analysis

Oberseminar Geometrie, Topologie und Analysis

H. Geiges, P. Konstantis, A. Lytchak, G. Marinescu

Wintersemester 2019/20

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

11.10.19 BGHK Seminar in Köln

18.10.19 Fernando Galaz-García (KIT)
Smooth 2-torus actions on the 5-sphere

Abstract: Isometric torus actions on closed, simply-connected Riemannian manifolds with positive and non-negative sectional curvature have been extensively studied. In dimension 5, Rong proved that a closed, simply-connected Riemannian 5-manifold M with positive sectional curvature and an isometric action of the 2-torus must be diffeomorphic to the 5-sphere. In the case where M is assumed to have non-negative sectional curvature, Searle and I obtained a classification up to diffeomorphism of such manifolds. With these classification results in place, a next natural step is to classify all possible actions of the 2-torus on a given manifold. In this talk I will discuss the general equivariant classification of smooth 2-torus actions on the 5-sphere, as well as the equivariant classification of isometric 2-torus actions on a 5-sphere with a Riemannian metric with positive sectional curvature. This is joint work with Diego Corro and Martin Kerin.

25.10.19 BGHK Seminar in Heidelberg

15./16.11.19 Geometric Dynamics Days in Aachen

22.11.19 BGHK Seminar in Bochum

17.1.20 Jason Lotay (Oxford)
Minimal surfaces, mean curvature flow and the Gibbons-Hawking ansatz

Abstract: The Gibbons-Hawking ansatz is a powerful method for constructing a large family of hyperkähler 4-manifolds (which are thus Ricci-flat), which appears in a variety of contexts in mathematics and theoretical physics. I will describe work in progress to understand the theory of minimal surfaces and mean curvature flow in these 4-manifolds. In particular, I will explain a proof of a version of the Thomas-Yau Conjecture in Lagrangian mean curvature flow in this setting. This is joint work with G. Oliveira.

24.1.20 Panagiotis Polymerakis (MPI Bonn)
Liouville properties of covering spaces

Abstract see here

31.1.20 Davide Barilari (Paris 7)
On the Brunn-Minkowski inequality

Abstract: The classical Brunn-Minkovski inequality in the Euclidean space generalizes to Riemannian manifolds with Ricci curvature bounded from below. Indeed this inequality can be used to define the notion of "Ricci curvature bounded from below" for more general metric spaces. A class of spaces which do not satisfy this more general definition is the one of sub-Riemannian manifolds: these can be seen as a limit of Riemannian manifolds having Ricci curvature that is unbounded, whose prototype is the Heisenberg group. In this talk I will discuss about the validity of a Brunn-Minkovsky type inequality in this setting and, if time permits, the related comparison theorems. [Joint work with Luca Rizzi].

H. Geiges, 5.4.02


11.10.19	BGHK Seminar in Köln
18.10.19	Fernando Galaz-García (KIT) Smooth 2-torus actions on the 5-sphere Abstract: Isometric torus actions on closed, simply-connected Riemannian manifolds with positive and non-negative sectional curvature have been extensively studied. In dimension 5, Rong proved that a closed, simply-connected Riemannian 5-manifold M with positive sectional curvature and an isometric action of the 2-torus must be diffeomorphic to the 5-sphere. In the case where M is assumed to have non-negative sectional curvature, Searle and I obtained a classification up to diffeomorphism of such manifolds. With these classification results in place, a next natural step is to classify all possible actions of the 2-torus on a given manifold. In this talk I will discuss the general equivariant classification of smooth 2-torus actions on the 5-sphere, as well as the equivariant classification of isometric 2-torus actions on a 5-sphere with a Riemannian metric with positive sectional curvature. This is joint work with Diego Corro and Martin Kerin.
25.10.19	BGHK Seminar in Heidelberg
15./16.11.19	Geometric Dynamics Days in Aachen
22.11.19	BGHK Seminar in Bochum
17.1.20	Jason Lotay (Oxford) Minimal surfaces, mean curvature flow and the Gibbons-Hawking ansatz Abstract: The Gibbons-Hawking ansatz is a powerful method for constructing a large family of hyperkähler 4-manifolds (which are thus Ricci-flat), which appears in a variety of contexts in mathematics and theoretical physics. I will describe work in progress to understand the theory of minimal surfaces and mean curvature flow in these 4-manifolds. In particular, I will explain a proof of a version of the Thomas-Yau Conjecture in Lagrangian mean curvature flow in this setting. This is joint work with G. Oliveira.
24.1.20	Panagiotis Polymerakis (MPI Bonn) Liouville properties of covering spaces Abstract see here
31.1.20	Davide Barilari (Paris 7) On the Brunn-Minkowski inequality Abstract: The classical Brunn-Minkovski inequality in the Euclidean space generalizes to Riemannian manifolds with Ricci curvature bounded from below. Indeed this inequality can be used to define the notion of "Ricci curvature bounded from below" for more general metric spaces. A class of spaces which do not satisfy this more general definition is the one of sub-Riemannian manifolds: these can be seen as a limit of Riemannian manifolds having Ricci curvature that is unbounded, whose prototype is the Heisenberg group. In this talk I will discuss about the validity of a Brunn-Minkovsky type inequality in this setting and, if time permits, the related comparison theorems. [Joint work with Luca Rizzi].