135 problems

Determine whether every compact foliated band with positive leafwise scalar curvature satisfies the foliated analogue of Gromov's sharp band-width estimate, bounding its leafwise w…

The conjecture asserts a quantitative width bound for foliated positive-scalar-curvature geometry: under the appropriate completeness, foliation, boundary, index, rank, and leafwis…

Let $M_g^2$ be a surface in ${\mathbb T}^3$, let $B$ be a direction, and let ${\cal F}_B(M_g^2)$ denote the corresponding foliation or section system. Call ${\cal F}_B(M_g^2)$ chao…

Molino's conjecture. The partition $\overline{\mathcal{F}}$ given by the closures of the leaves of $\mathcal{F}$ is again a singular Riemannian foliation.

Let $(M,\gamma)$ be a taut sutured manifold, meaning a sutured manifold satisfying the tautness condition, and suppose that $\chi(M)<0$. A finite-sheeted cover is a covering … with…

Let $(M,\mathcal{F})$ be a foliation on a compact manifold, let $BG$ be the classifying space of its holonomy groupoid, and let $\mathcal{L}^{lf}$ be the cosheaf assigning to each…

Let $X$ be a projective rationally connected manifold and let $\mathcal{F}$ be a regular foliation on $X$. Touzet's conjecture. The foliation $\mathcal{F}$ is induced by a smooth m…

Let $(G,g)$ be a semi-Riemannian Lie group with a subgroup $K$ generating a left-invariant conformal foliation $\mathcal F$ on $G$ of codimension two. Semisimple subgroup minimalit…

Let $M$ be a closed simply connected manifold admitting a Riemannian metric of nonnegative sectional curvature. Grove's conjecture. Then $M$ has a DDBD structure, that is, an isopa…

The smoothness conjecture. If $\operatorname{codim}\mathcal{F}>0$ and the leaves are dense, then

A volume-preserving partially hyperbolic diffeomorphism is a diffeomorphism preserving a volume measure and admitting a partially hyperbolic invariant splitting; when its center fo…

Let ${\mathcal F}$ be an arbitrary foliation on a compact Riemannian manifold. In the above notation, let $\Delta_F$ and $\Delta_H$ denote the leafwise and transverse Laplacians, r…

Let $\mathcal L$ be a lamination arising from a group action. A fidelity of $\mathcal L$ is a germ map into a path-connected space whose induced map on fundamental germ groupoids i…

Let $\chi$ be a reduced analytic vector field on a real analytic manifold $M$ without boundary, and let $U\subset M$ be relatively compact. Write $L_\chi|_U$ for the line field ind…

Hypertight-contact-structure conjecture. Every closed, orientable manifold that carries a taut foliation also carries a hypertight contact structure.

A solenoid by holomorphic curves is a compact solenoid in the complex projective plane whose leaves are holomorphic curves. Holomorphic-solenoid conjecture. There exists a compact…

Let $M^4$ be a four-manifold with a taut foliation ${\mathcal F}$, and let $J_{\mathcal F}$ be an almost-complex structure induced by ${\mathcal F}$. For an embedded surface $S$ in…

Let $M^4$ be a $4$-manifold, let ${\mathcal F}$ be a taut foliation on $M$, and let $J_{\mathcal F}$ be an almost-complex structure induced by ${\mathcal F}$. For an embedded surfa…

Let $X$ be a compact manifold with a one-codimensional foliation ${\mathcal F}$ and an ${\mathcal F}$-compatible flow $\phi$. Assume that the fixed points and periodic orbits are n…

Characteristic-class conjecture. The characteristic classes of codimension-$p$ foliations on $X$ coincide with the elements of

A spacetime is maximally extended if it admits no further extension, and it is globally hyperbolic if it has a Cauchy surface. A constant mean curvature (CMC) hypersurface is a spa…

Let $\Lambda^s$ and $\Lambda^u$ denote the stable and unstable foliations of a pseudo-Anosov flow $\varphi$. A branching orbit is a closed orbit that lies on a leaf of $\Lambda^s$ or…

Conjecture. Every codimension-one foliation on $X$ either admits a singular transversely projective structure, or is the pull-back, by a rational map, of a foliation on a complex s…

Let $d$ be a positive integer, and let $\Gamma\subset [0,1]$ be a DCC set of rational numbers whose closure is contained in $\mathbb Q$. A projective $\mathbb R$-complementary norm…

McKernan's ACC conjecture. There exists an ACC set $\Gamma\subset [0,1]$ depending only on $d$ such that, for every such $\mathcal{F}$ on $X$, the threshold…