16 problems

Completeness conjecture. There exists such a metric $G^k$ for which: (a) $(\operatorname{Imm}^k(M,N),G^k)$ is metrically complete; (b) $(\operatorname{Imm}^l(M,N),G^k)$ is geodesic…

Mumford's completeness conjecture. Higher-order Sobolev metrics on spaces of immersed surfaces are expected to be complete.

Michor–Mumford conjecture. For every $k\geq 0$, the map

Blowup conjecture. Blowup of solutions occurs for every metric of order $s<\frac32$.

Let $M$ be a manifold of dimension at least $3$, and let $\operatorname{Diff}_\mu(M)$ denote its group of volume-preserving diffeomorphisms equipped with the right-invariant $H^s$…

For a group of diffeomorphisms, define the distance threshold by … the smoothness threshold by … and the Fredholm threshold by … Here $d_s$ is the $H^s$ geodesic distance,…

Let $M$ be a general manifold, and let $\beta_s$ denote the right-invariant $H^s$ metric on the diffeomorphism group of $M$. The critical-index extension conjecture. The known vani…

For the geodesic-distance results established in the paper, including vanishing of the $H^{1/2}$ distance on diffeomorphism groups and vanishing of the $H^{-1/2}$ distance on group…

Let the group of volume-preserving diffeomorphisms carry the right-invariant $H^{-1/2}$ metric. Washabaugh's conjecture. The geodesic distance of this metric is degenerate: there a…

Let $\mathcal{M}$ be a Riemannian manifold, and let $\operatorname{Diff}_\text{c}(\mathcal{M})$ be its group of compactly supported diffeomorphisms equipped with a right-invariant…

Let $\mathcal{D}_\mu(\mathbb{T}^2)$ be the volumorphism group of the flat torus equipped with the $\dot{H}^{-1/2}$ metric. Its sectional curvature is unbounded of both signs. Micho…

Positivity conjecture. The $H^{1/2}$-metric has positive sectional curvature.

Strongness conjecture. Sobolev metrics of order $n\geq 3$ should be strong Riemannian metrics on $\mathcal I^n(M,\mathbb R^d)$.

Let $M$ be the underlying manifold and let $s>\frac12$. Consider the corresponding right-invariant Sobolev metric on the diffeomorphism group. Positivity conjecture. The geodesic d…

Let $M$ be an arbitrary Riemannian manifold of bounded geometry, and let $\operatorname{Diff}_c(M)$ denote its group of compactly supported diffeomorphisms. For…

Let $M=\mathbb{R}$ and let $s=\frac12$. Consider the right-invariant Sobolev metric of order $s$ on $\operatorname{Diff}_c(M)$. Vanishing-distance conjecture. The geodesic distance…