Topology
Using Ricci flow to study spaces of metrics and diffeomorphism groups.
Concept
A topological payoff of flowing through singularities. For a spherical space form , the inclusion of the isometry group into the diffeomorphism group
is a homotopy equivalence. Hatcher proved the case by hand in 1983; Bamler–Kleiner proved the general case by running Ricci flow on families of metrics and using uniqueness of singular flows to contract the space of metrics onto the round one.
On arXiv
The Swampland Distance Conjecture postulates the emergence of an infinite tower of massless states when approaching infinite-distance points in moduli space. However, most string backgrounds are supported by fluxes, and therefore depart from the purely geometric paradigm. This fact requires an extension of the Swampland conjectures to scalar field spaces with non-trivial potentials, rather than just moduli spaces. To address this task, we utilise geometric flows, in particular generalised Ricci flow, to probe the associated scalar field spaces. Considering internal spaces supported by three-form fluxes, we first show that the distance defined in terms of the Perelman entropy functional needs to be refined in order to encompass fluxes. Doing so, we extend the Ricci Flow Conjecture to include Kalb-Ramond flux besides the metric and the dilaton field. This allows us to probe infinite-distance points within these scalar field spaces in a purely geometric way. We subsequently construct a geometric flow for internal manifolds supported by Ramond-Ramond fluxes and discuss its role in the Ricci Flow Conjecture. Our analysis suggests that in the presence of fluxes the Distance Conjecture might be better characterised in terms of a cost function on the space of metrics, rather than a genuine distance.
We give a new proof of the Smale conjecture for and all lens spaces using minimal surfaces and min-max theory. For , the conjecture was first proved in 2019 by Bamler-Kleiner using Ricci flow.