Research Seminar on Topology (Oberseminar Topologie)
Mondays 17-18 in 0.05 (PER 23)

Since the introduction of Ricci flow by Hamilton in 1982, it has been a fundamental question to understand the evolution of metrics and their curvature conditions under the flow. While positive scalar curvature and 2-positive curvature operator are preserved in all dimensions, there exist infinitely many dimensions where certain curvature conditions lying in between are not preserved. In this talk, I first recall some basics about homogeneous Ricci flow. Then, I present joint works with David González-Álvaro in which we examine metrics with different curvature conditions on various homogeneous spaces and discuss the evolution of their metrics under the Ricci flow.

The twisted suspension of a manifold can be seen as a smooth analogue of the classical suspension operation for topological spaces. Its construction is motivated from the spinning operation in knot theory and it is obtained by surgery on a fibre of a principal circle bundle over the given manifold. In this talk I will show that Riemannian metrics of positive Ricci curvature can be lifted along twisted suspensions. As application we obtain first examples of simply-connected manifolds of positive Ricci curvature with maximal symmetry rank in any dimension, and we obtain new examples of (rational) homology spheres with a Riemannian metric of positive Ricci curvature.

The notion of $m$-intermediate curvature interpolates between Ricci curvature and scalar curvature. In this talk we describe extensions of classical results by Bonnet-Myers and Schoen-Yau to the setting of $m$-intermediate curvature: A non-existence result for metrics of positive $m$-intermediate curvature on manifolds with topology $N^n=T^m$×$S^{n-m}$; a gluing result for manifolds with $m$-convex boundary; inheritance of spectral positivity along stable minimal hypersurfaces, and estimates for the $m$-diameter for uniform positive lower bounds. This talk is partially based on joint work with Simon Brendle and Sven Hirsch, and joint work with Aaron Chow and Jingbo Wan.

In this talk we will discuss the construction of ${\mathrm{Ric}}_2$>0 metrics on closed manifolds of dimensions 10, 11, 12, 13 and 14. These include $S^6$×$S^7$, $S^7$×$S^7$ and infinitely many homeomorphism types in dimension 13. The main new ingredient is a notion for homogeneous bundles which generalizes the concept of fatness and, together with other assumptions, ensures the existence of ${\mathrm{Ric}}_2\mathrm{>0}$ metrics on the total space of the bundle. Further examples are obtained by considering biquotients of the corresponding homogeneous spaces. This is joint work with Jason DeVito, Miguel Domínguez-Vázquez and Alberto Rodríguez-Vázquez.

In this talk we shall review a few of the many existing notions of curvature in Riemannian and metric geometry, together with some of the main achievements and current lines of research. We will put special focus on manifolds of positive sectional curvature and those of positive Ricci curvature, and also on certain curvature conditions which interpolate between them.

This talk centres around the study of sequences of minimal surfaces in a three manifold. So far, an in-depth characterisation of such sequences has been carried out in the case where the surfaces considered are discs, both in regards to the curvature blow-up points and the limit surfaces that can arise. In order to study the topology of the limit leaves of sequences of properly embedded minimal discs, Bernstein and Tinaglia first introduced the concept of the simple lift property, since such limit leaves satisfy this property. They proved that an embedded surface Σ ⊂ Ω with the simple lift property must have genus zero, if Ω is an orientable three-manifold satisfying certain geometric conditions. During my PhD, I generalised this result by considering an arbitrary orientable three-manifold Ω and proving that one is still able to restrict the topology of an arbitrary surface Σ ⊂ Ω with the simple lift property. In this talk, I will present which topological results can be extended in the case of sequences of minimal surfaces of arbitrary finite genus.

The Bakry-Émery Ricci tensor is a generalization of the classical Ricci tensor to the setting of weighted Riemannian manifolds, i.e. Riemannian manifolds whose Riemannian volume forms are weighted by a smooth function. While it was defined by Bakry and Émery in the context of diffusion processes, it also appears naturally in the study of Ricci flow and collapsed Ricci limit spaces. In analogy to important open problems in the Riemannian case, we will consider the question of which manifolds admit a weighted Riemannian metric of positive Bakry-Émery Ricci curvature. We will show that surgery techniques that are used to construct examples in the Riemannian case can be extended and improved in the weighted case. As applications we construct new examples of such manifolds in dimension 5. This is joint work with Francesca Tripaldi.