14 problems

Cut-time and cut-locus conjecture. The cut time should be

Dimension-free cut-locus structure conjecture. The cut locus has the form

Let $\mathbb F_4$ be the free step-2 rank-four Carnot group, and let $\mathcal H$ be the union of all strict Carnot subgroups. Equivalently, … Let…

Let $\mathbb F_n$ be the free step-2 rank-$n$ Carnot group, and let $\operatorname{Cut}(\mathbb F_n)$ denote its cut locus. Let $C_n\subset\mathbb F_n$ be the set defined in the so…

Let $\pi:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ be the quotient map, let … and set $\tilde{X}(d)=\pi^{-1}(X(d))$. Here $\mathbb{Z}_d^{\star(n+1)}$ denotes the $(n+1)$-fold topological…

Let $({\mathbb{S}}^2,g)$ be a 2-sphere of revolution with reflective symmetry, where the metric has the form … and the Gaussian curvature is considered along meridians. Gaussian-cu…

Existence conjecture. We conjecture that such examples exist, leaving their concrete construction for future research.

Let $P(\beta)$ be the period function in $G_\alpha$, with $\beta$ the parameter used to specify the initial tangent vector. The period-function conjecture. … and … The source estab…

Let $(M,g)$ be a compact and simply connected Riemannian $n$-manifold, and let $x\in M$. Denote by $\widetilde C(x)\subset T_xM$ the cut locus and by $\widetilde J(x)\subset T_xM$…

Let $\mathbb{G}_k$ be the free step-two Carnot group of rank $k$, identified with $\mathbb{R}^k\oplus\wedge^2\mathbb{R}^k$, and let $\mathrm{Cut}_0(\mathbb{G}_k)$ denote its cut lo…

Consider the sub-Riemannian problem on the Lie group $\mathrm{SH}(2)$, with its exponential mapping and the discrete symmetry group acting on that mapping. Let the cut time be the…

Let $M$ be an $n$-dimensional Riemannian manifold, let $p\in M$, and consider the cut locus of $p$. An optimal conjugate point is a conjugate point of $p$ lying on a minimizing geo…

Let $M$ be a manifold whose regularity is insufficiently smooth, and let the cut locus of a point or submanifold in $M$ be equipped with its Hausdorff dimension. Non-integer Hausdo…

Cut-locus mass conjecture. The cut locus $C(\mu)$ cannot carry mass if $\mu$ is a local minimizer.