Smale conjecture & diffeomorphisms

We study the topology of the space of all smooth asymptotically stable vector fields on ${\mathbb{R}}^n$, as well as the space of all proper smooth Lyapunov functions for such vector fields. We prove that both spaces are path-connected and simply connected when $n\ne 4,5$ and weakly contractible when $n\le 3$. Moreover, both spaces have the weak homotopy type of the nonlinear Grassmannian of submanifolds of ${\mathbb{R}}^n$ diffeomorphic to the $n$-disc. The proofs rely on Lyapunov theory and differential topology, such as the work of Smale and Perelman on the generalized Poincaré conjecture and results of Smale, Cerf, and Hatcher on the topology of diffeomorphism groups of discs. Applications include a partial answer to a question of Conley, a parametric Hartman-Grobman theorem for nonyperbolic but asymptotically stable equilibria, and a parametric Morse lemma for degenerate minima. We also study the related topics of hyperbolic equilibria, Morse minima, and relative homotopy groups of the space of asymptotically stable vector fields inside the space of those vanishing at a single point.

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.

In this article, we recapture the Smale conjecture on a Sasakian $3$-sphere via the Legendrian mean curvature flow. More precisely, we deform the area-preserving contactomorphism (symplectomorphism) of Sasakian $3$-spheres to an isometry via the Legendrian mean curvature flow on the Legendrian graph in ${\mathbb{S}}^2\times {\mathbb{S}}^3$. By using the monotonicity formula and blow-up analysis, we obtain the minimal Legendrian graph in ${\mathbb{S}}^2\times {\mathbb{S}}^3$. Finally, we will address the rigidity theorem of $2$-dimensional Legendrian self-shrinkers in ${\mathbb{R}}^5$. We are able to reconstruct the Harvey-Lawson special Lagrangian cone in ${\mathbb{C}}^3$ from this Legendrian self-shrinker. The partial classification is also provided if the squared norm of the second fundamental form is constant.

We give a new proof of the Smale conjecture for ${\mathbb{RP}}^3$ and all lens spaces using minimal surfaces and min-max theory. For ${\mathbb{RP}}^3$, the conjecture was first proved in 2019 by Bamler-Kleiner using Ricci flow.

A generalized metric on a manifold $M$, i.e., a pair $(g,H)$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, is a fixed point of the generalized Ricci flow if and only if $(g,H)$ is Bismut Ricci flat: $H$ is $g$-harmonic and $ric(g)=\frac{1}{4}{H}_g^2$. On any homogeneous space $M=G/K$, where $G=G_1\times G_2$ is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat $G$-invariant generalized metric, which is proved to be unique among a $4$-parameter space of metrics in many cases, including when $K$ is neither abelian nor semisimple. On the other hand, if $K$ is simple and the standard metric is Einstein on both $G_1/{\pi }_1(K)$ and $G_2/{\pi }_2(K)$, we give a one-parameter family of Bismut Ricci flat $G$-invariant generalized metrics on $G/K$ and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form $M=G\times G/\Delta K$ and for $M^{35}=SO(8)\times SO(7)/G_2$. A Corrigendum has been added in Appendix A.

We show that $S^2\times S^2$ is isolated as a shrinking Ricci soliton in the space of metrics, up to scaling and diffeomorphism. We also prove the same rigidity for $S^2\times N$, where $N$ belongs to a certain class of closed Einstein manifolds. These results are the Ricci flow analogues of our results for Clifford-type shrinking solitons for the mean curvature flow.