On the Geometry of Ends of Ricci Shrinkers

In this thesis we study a particular type of Riemannian manifolds, namely gradient shrinking Ricci solitons, or Ricci shrinkers for short. Besides being generalisations of positive Einstein manifolds, Ricci shrinkers play a fundamental role in Ricci flow theory, modeling the finite-time singularities of the flow. While Ricci shrinkers are fully classified in dimensions two and three, in higher dimensions such a classification is far from being complete, and it is customary to study instead their general geometric properties and moduli spaces. In this work we deal with the so-called ends of a Ricci shrinker, which are, roughly speaking, its connected components at infinity. After introducing the necessary theory in Chapter 1, in Chapter 2 we give, under assumptions on the Perelman entropy or on the potential function of the shrinker, upper bounds on the number of ends a Ricci shrinker can have, and prove that this number is a lower semicontinuous function along any converging sequence of pointed Ricci shrinkers. This is done both for general and for asymptotically conical ends, a particular class of ends with an interesting geometrical characterisation. We also prove that if a Ricci shrinker contains a subset that is near to a cone in a precise sense, then there exists an asymptotically conical end containing it. Finally, we apply these results to derive some information about the boundary points of moduli spaces of Ricci shrinkers. From the technical point of view, we need to introduce and work with singular spaces and singular Ricci shrinkers, which are metric spaces with a regular-singular decomposition, and with a related weak notion of convergence. Chapter 3 is dedicated to the study of blow-up sequences of Ricci shrinkers without global curvature assumptions, still having in mind possible applications to the geometry of ends. In this context, the classical convergence theory for Ricci flows is no longer adequate, and we need to work with the F-convergence theory recently introduced by Bamler. This theory is expressed in terms of metric flow pairs, i.e. metric flows equipped with a conjugate heat flow, generalisations of Ricci flows and of solutions of the conjugate heat equation in the context of metric spaces, respectively. First of all, we prove the existence of a heat kernel function based at the singular time of the Ricci flow induced by a Ricci shrinker and derive its main properties. This allows us to convert a Ricci shrinker into a metric flow pair defined up to the singular time, making it possible to treat blow-up sequences in this framework. We then prove that any sequence of metric flow pairs associated with Ricci shrinkers F-converges to a limit metric flow, and, under an entropy lower bound on the sequence, we derive some structural properties of the limit space. In the case of blow-up sequences of Ricci shrinkers, we obtain convergence towards a continuous metric soliton, where metric solitons are the analogue of Ricci shrinkers in Bamler’s theory. In this sense, our results extend what is known for Type I Ricci flows. At this point, assuming a local Type I bound on the scalar curvature at the basepoint of the blow-up sequence, we prove that the limit metric flow splits a line. Due to their self-similarity, we can study the geometry at infinity of Ricci shrinkers using blow-up sequences, thus obtaining that any end of a Ricci shrinker splits a line.