Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

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We investigate Riemannian metrics on compact, non-compact and singular manifolds. On the one hand, important questions in differential geometry concern the existence and construction of (complete) metrics with specific curvature properties. On the other hand, the solution of the existence question motivates the study of the space of all such metrics. In this project we will mainly be dealing with metrics with lower scalar, Ricci and sectional curvature bounds. The goal is an improved understanding of the structure and of the topological properties of spaces and moduli spaces of such metrics.