9 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.

1/2-conjecture. There are $2k+1$ Neumann–Poincaré eigenvalues whose sum is

Let $z_n$ denote the logarithmic fluctuation associated with the empirical Fréchet mean on a Riemannian manifold, and let the first moment (bias) be…

Local SRVT distance conjecture. For general homogeneous manifolds, at least locally, the geodesic distance of the pullback metric is given by the $L^2$ distance between the curves…

Euler sprays conjecture. For sufficiently small $\varepsilon>0$, Euler sprays can be constructed that hit $\Omega_1$ up to a set of measure zero.

Let the initial path be the path joining a circle to the same circle while passing through an ellipse in the middle, and let the constant path containing only circles be the corres…

The atrophy-constrained LDDMM optimal control problem uses a deformation velocity together with affine transformations, including noncompact components of the affine group, and its…

Let a distribution on a shape space be sufficiently concentrated, and let its Procrustean mean be defined in the usual sense. Stability conjecture. Procrustean means of sufficientl…