Spaces and Moduli Spaces of Riemannian Metrics

Abstract: Consider a smooth manifold with a Riemannian metric satisfying some sort of curvature constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasaki, etc. A natural question to study is then what the space of all such metrics does look like. Moreover, one can also pose this question for corresponding moduli spaces of metrics, i.e., quotients of the former by (suitable subgroups of) the diffeomorphism group of the manifold, acting by pulling back metrics.

These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties then provide the right means to measure 'how many' different metrics and geometries the given manifold actually does exhibit; but one can topologize and view those also in very different manners.

In my talk, I will report on some general results and open questions about spaces and moduli spaces of metrics with non-negative Ricci or sectional curvature as well as other lower curvature bounds on closed and open manifolds, and, in particular, also discuss broader non-traditional approaches from metric geometry and analysis to these objects and topics.

algebraic topologydifferential geometrymetric geometry

Audience: researchers in the topic

Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php