Mini-Workshop in Metric Geometry

Saturday Jun 24, 2017, 8:00 AM → 6:00 PM Europe/Berlin

Seminar Room 1.067 (Kollegiengebäude Mathematik (Building 20.30))

Mini-Workshop in Metric Geometry

Mini-Workshop in Metric Geometry in the framework of the Dual Year Mexico-Germany.

The workshop was organized with the support of Mexico's Embassy in Germany and of KIT's International Affairs Office.

Speakers

• Jesús Núñez-Zimbrón (University of California, Santa Barbara, USA).
• Oscar Palmas (Facultad de Ciencias, UNAM, Mexico).
• Raquel Perales (Instituto de Matemáticas, UNAM, Mexico).
• Pedro Solórzano (Instituto de Matemáticas, UNAM, Mexico).
• Pablo Suaréz Serrato (Instituto de Matemáticas, UNAM, Mexico).

Local Organizers

• Diego Corro
• Fernando Galaz-García
• Sebastian Grensing
• Wilderich Tuschmann

Background image courtesy of H.-J. Sommerfeld

1 Geometry of null hypersurfaces

Given a semi-Riemannian manifold $\bar M$ with metric $\bar g$, a submanifold $M$ of $\bar M$ is null if the restriction $g$ of $\bar g$ to $M$ is degenerate. These objects deserve some attention, because of their connection with some areas like the study of black holes and others. In general, some properties of $M$ cannot be studied with the usual differential geometric methods, but here we discuss the case of null hypersurfaces in Lorentzian space forms. This is a joint work with M. Navarro and D. Solis (UADY-Mexico)

Speaker: Prof. Oscar Palmas (Facultad de Ciencias, UNAM, Mexico)

2 A couple remarks on parallelism.

A submetry is a natural generalization of a riemannian submersion, which in turn is a natural generalization of an isometry. We will see how certain properties of riemannian submersions remain valid for submetries provided one is willing to relax their definitions. In particular, the notion of parallel translation along arbitrary paths has a natural generalization. Existence and uniqueness are no longer guaranteed. Mild assumptions yield existence. Uniqueness, on the other hand, is harder to subdue. We will discuss natural consequences of uniqueness.

Speaker: Prof. Pedro Solórzano

11:45 AM LUNCH

3 Alexandrov and Integral Current Spaces (with Jaramillo, Rajan, Searle and Siffert).

In this talk we will see that given a closed, orientable Alexandrov space (X,d) we will define an integral current T defined on X with weight equal to 1 and boundary of T equal to zero. In other words, this extends to Alexandrov Spaces the canonical current defined on a closed oriented Riemannian manifold. We will recall that non collapsing sequences of closed oriented compact Riemannian manifolds with a uniform diameter upper bound and a uniform lower Ricci curvature bound have subsequences that converge in both intrinsic flat and Gromov-Hausdorff sense to the same limit space. Thus, at the end of the talk we will study the intrinsic flat convergence of sequences of Alexandrov Spaces.

Speaker: Prof. Raquel Perales (Instituto de Matemáticas, UNAM, Mexico)

4 Equivariant geometry of Alexandrov 3-spaces.

Alexandrov spaces constitute a synthetic generalization of Riemannian manifolds with a lower bound on sectional curvature. They provide a natural metric setting to study notions of global Riemannian geometry, and therefore, a fundamental problem is that of extending to Alexandrov spaces what is known for Riemannian manifolds. As in the Riemannian case, one may investigate Alexandrov spaces via their symmetries. Since the isometry group of a compact Alexandrov space is a compact Lie group, this point of view naturally leads to the study of isometric Lie group actions on Alexandrov spaces. In this context, a natural measure of the complexity of an action is the cohomogeneity, i.e. the dimension of the orbit space. Berestovskii showed that finite-dimensional homogeneous Alexandrov spaces actually are Riemannian manifolds. Galaz-Garcia and Searle studied Alexandrov spaces of cohomogeneity 1 and classified them in dimensions at most 4. In this talk I will present an equivariant and topological classification of closed Alexandrov spaces of dimension 3 admitting isometric actions of cohomogeneity 2 as well as a generalization to the case of isometric local circle actions. As an application of this result I will talk about some aspects of the geometry (in the sense of Thurston) of Alexandrov 3-spaces. The results presented here are joint work with Fernando Galaz-García and Luis Guijarro.

Speaker: Prof. Jesus Núñez-Zimbrón ((University of California, Santa Barbara, USA)

3:45 PM COFFEE

5 On the topology and geometry of higher graph manifold

Our understanding of $3$--manifolds has illuminated two distinct classes of importance; hyperbolic manifolds and graphmanifolds. These are by now considered the basic blocks featured in the geometrisation program of Thurston, famously consolidated by Perelman. From one perspective graph manifolds are exactly the manifolds that {\it collapse}, in the sense that they admit a family of smooth metrics whose volumes tend to zero while their sectional curvatures remain bounded. Historically the term {\it graph manifold} was introduced by Waldhausen in the 1960's. It highlighted the fact that the fundamental group can be described as a graph of groups and that the manifolds were built up from fundamental pieces that are (heuristically) described in terms of circle bundles over 2-orbifolds. Recently Frigerio, Lafont and Sisto proposed a family of generalised graph manifolds; products of k--tori with hyperbolic (n-k)--manifolds with truncated cusps are glued along their common n--toral boundaries. They explored multiple topological aspects of this family and raised some questions. For example, in their definition k is allowed to equal zero, so that one subfamily is made up of hyperbolic manifolds glued along truncated cusps. They asked if the minimal volume of such manifolds is achieved by the sum of the hyperbolic volumes of the pieces. Together with Chris Connell we answered this question positively. In so doing we realized that a natural family we termed higher graph manifolds could be defined; bundles of infranilpotent manifolds over negatively curved bases are glued along boundaries (when possible). This family further extends the one proposed by Frigerio, Lafont and Sisto. We first characterize the higher graph manifolds that admit volume collapse, by explicitly constructing sequences of metrics of bounded curvature whose volume collapses (this builds on earlier work by Fukaya). Various results about the simplicial volume and volume entropy of this family are calculated. Then we exploit the graph structure of the fundamental group to show that these manifolds obey the coarse Baum-Connes conjecture, have finite asymptotic dimension and do not admit metrics of positive scalar curvature. Finally we use several of the produced results to prove that when the infranilmanifold fibre has positive dimension the Yamabe invariant vanishes. Further, in joint work with Noé Bárcenas and Daniel Juan Pineda we proved that the Borel conjecture also holds for higher graph manifolds.

Speaker: Prof. Pablo Suárez Serrato (Instituto de Matemáticas UNAM, Ciudad de México)