UC San Diego

Ricci flow is a powerful and fundamentally innovative tool in the field of geometric analysis introduced by Richard Hamilton [Ha82] in 1982. Many longstanding geometric and topological problems have been solved using Ricci flow. For example, the Poincaré conjecture, Thurston’s geometrization conjecture, the $1/4$-pinched differentiable sphere theorem, the generalized Smale conjecture, and so on. In the seminal works of Hamilton and Perelman, it is crucial to understand at least the qualitative behavior of the singularities so that Ricci flow can be continued via surgeries. To understand the singularity formation, it is desirable to classify singularity models, which are the blow-up limits along sequences of space-time points with curvature tending to infinity, or at least to understand them qualitatively well enough for topological applications. However, as compared to dimension 3, the geometry becomes drastically more complicated starting from dimension 4, and so does the singularity analysis of Ricci flow. Recently, Richard Bamler in [Bam20a,Bam20b,Bam20c] established a groundbreaking theory for the weak limits (his F-limits) of Ricci flows on closed manifolds in higher dimensions. This theory will be fundamentally important in the study of higher-dimensional Ricci flow singularities. All the known examples so far suggest that Ricci flow singularity models should be mostly shrinking or steady Ricci solitons. These are the self-similar ancient solutions to Ricci flow, and they can be viewed as generalized Einstein manifolds. Thus, it is vitally important to study shrinking and steady Ricci solitons that arise as singularity models.

In the dissertation, we shall survey some recent results on the geometry in the large of singularity models or more general noncollapsed ancient solutions of Ricci flow, which were jointly investigated by collaborators and the author. We shall streamline some proofs and also present some new unpublished findings. There are two fundamental notions of space-time blow-downs for ancient flows: asymptotic shrinking solitons by Perelman [Per02] and tangent flows at infinity by Bamler [Bam20c]. We will show that the two notions coincide, which is the main theorem of [CMZ21a]. We will present that for ancient flows, various entropy quantities introduced mainly by Perelman converge to those of the tangent flows at infinity, which were proved in [MZ21,CMZ21a,CMZ21b,CMZ21d]. It will also be demonstrated that how tangent flows at infinity determine the geometry in the large of steady solitons in dimension 4, which is the main theorem of [BCDMZ]. Moreover, we will present an optimal volume growth estimate for noncollapsed steady solitons in all dimensions, which is the main result of [BCMZ21].