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We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications.
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In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In addition, for Ricci harmonic flow, we use the monotonicity of functional $ u_alpha$ to show the connection between finite-time singularity and shrinking Ricci harmonic soliton. At last, we explore the property of ancient solutions for Ricci harmonic flow.
We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelmans conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three manifolds --- in particular to the Generalized Smale Conjecture --- which will appear in a subsequent paper.
We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangen- tial to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metrics with sufficiently regular heat kernels. On an appropriate non- singular open region, we provide a family of metric tensors evolving in time and provide a regularity theory for this flow in terms of the regularity of the heat kernel. When the rough metric induces a metric measure space satisfying a Riemannian Curvature Dimension condition, we demonstrate that the distance induced by the flow is identical to the evolving distance metric defined by Gigli and Mantegazza on appropriate admissible points. Consequently, we demonstrate that a smooth compact manifold with a finite number of geometric conical singularities remains a smooth manifold with a smooth metric away from the cone points for all future times. Moreover, we show that the distance induced by the evolving metric tensor agrees with the flow of RCD(K, N) spaces defined by Gigli-Mantegazza.
In this paper we apply techniques from optimal transport to study the neckpinch examples of Angenent-Knopf which arise through the Ricci flow on $mathbb{S}^{n+1}$. In particular, we recover their proof of single-point pinching along the flow. Using the methods of optimal transportation, we are able to remove the assumption of reflection symmetry for the metric. Our argument relies on the heuristic for weak Ricci flow proposed by McCann-Topping which characterizes super solutions of the Ricci flow by the contractivity of diffusions.
Associated to each finite dimensional linear representation of a group $G$, there is a vector bundle over the classifying space $BG$. We introduce a framework for studying this construction in the context of infinite discrete groups, taking into account the topology of representation spaces. This involves studying the homotopy group completion of the topological monoid formed by all unitary (or general linear) representations of $G$, under the monoid operation given by block sum. In order to work effectively with this object, we prove a general result showing that for certain homotopy commutative topological monoids $M$, the homotopy groups of $Omega BM$ can be described explicitly in terms of unbased homotopy classes of maps from spheres into $M$. Several applications are developed. We relate our constructions to the Novikov conjecture; we show that the space of flat unitary connections over the 3-dimensional Heisenberg manifold has extremely large homotopy groups; and for groups that satisfy Kazhdans property (T) and admit a finite classifying space, we show that the reduced $K$-theory class associated to a spherical family of finite dimensional unitary representations is always torsion.
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