26 problems

Let $M$ be a compact transverse Fano Sasakian manifold of dimension five, and let its Sasaki–Ricci flow be a solution of the flow equation referred to in the source. A shrinking Sa…

Let $S$ be a Sasaki–Einstein manifold. Cheeger–Tian's conjecture. The manifold $S$ is quasi-regular. The conjecture was refuted by irregular Sasaki–Einstein examples constructed la…

Let $M$ be a complete noncompact Sasakian $(2n+1)$-manifold with positive pseudohermitian bisectional curvature. Let $\mathbf{H}_n=\mathbb{C}^n\times\mathbb{R}$ be the standard…

Let $M$ be a Carnot–Carathéodory complete Sasakian manifold with pseudohermitian Ricci tensor $\rho$, and let $(\varphi,\xi,\eta,g)$ be its contact metric structure. For some…

An odd-dimensional sphere is the standard sphere $S^{2n+1}$, and a parallelizable manifold is a manifold with trivial tangent bundle. A Sasakian-Einstein metric is a Sasakian metri…

A homotopy sphere is a smooth manifold homotopy equivalent to a sphere, and a Sasakian-Einstein metric is a Sasakian metric whose associated Riemannian metric is Einstein. The prec…

Let $(M,\xi_{0},\eta_{0},\Phi_{0},g_{0},\omega_{0})$ be a compact quasi-regular Sasakian $5$-manifold with foliation quotient singularities of Type I. Suppose that the associated n…

Sasakian uniformization conjecture. Then $M$ is CR biholomorphic to the standard Heisenberg group $\mathbb{H}_{n}$.

Let $f$ be a ${\bf w}$-homogeneous polynomial defining a Calabi-Yau $3$-fold, and let the associated link have Sasakian Hodge number $h_S^{2,1}$, while the Calabi-Yau $3$-fold has…

Let $f,g$ be weighted homogeneous polynomials on $\mathbb{C}^n$ of the same degree $d$, with weight vectors ${\bf w}_f$ and ${\bf w}_g$. Two such polynomials are weakly R-equivalen…

In a closed $5$-dimensional Sasakian manifold $N^5$, consider a solution of the weak Hamiltonian-minimal equation arising for a continuous conformal Legendrian map from a closed Ri…

Let $S$ be a Sasaki manifold, let $a(S)$ be its normalized volume, and let the rank of $S$ be the dimension of the closure of a generic Reeb orbit. Martelli–Sparks–Yau's conjecture…

Boyer–Galicki–Kollár's conjecture. Every parallelizable exotic sphere admits a Sasakian structure.

Let an $n$-dimensional toric Gorenstein singularity be associated with a polytope $\Delta_{n-1}$, either reflexive or non-reflexive. Let $Y$ be the associated Sasaki–Einstein base…

An iterated $S^3$ join in the Gorenstein case is a smooth manifold of odd dimension. Existence conjecture. In the Gorenstein case, the iterated $S^3$ joins admit smooth Sasaki–Eins…

Let $M$ be a complete noncompact Sasakian $(2n+1)$-manifold with nonnegative pseudohermitian bisectional curvature, and let $\mathcal{O}_p^{CR}(M)$ denote the ring of all CR-holomo…

Let $M$ be a compact Sasakian $7$-manifold with positive transverse scalar curvature, and let ${\bf E}\to M$ be a Sasakian $G$-bundle. Let $A$ be an irreducible selfdual contact in…

Let $(M,g)$ be a $3$-Sasakian $7$-manifold, with Sasakian structures satisfying the quaternionic relations … Let $\mathcal{M}^\ast$ be the moduli space under consideration. The tra…

Let $(M^3,J,\theta(0))$ be a closed CR manifold with pseudo-Einstein contact form $\theta(0)$. Suppose that its Tanaka-Webster curvature $W(0)$ is positive and constant, and that ……

Let $S$ be a compact $\eta$-Einstein Sasakian manifold that admits a Sasakian immersion into a sphere. For a simply-connected compact homogeneous Hodge manifold $M$, let…

Boyer–Galicki's odd-dimensional Goldberg conjecture. Every compact Einstein $K$-contact manifold is Sasakian.

Nonexistence conjecture. There does not exist such an $X$ containing $b_2(X)$ disjoint complex curves all of genus $g\geq 1$.

Klebanov–Witten conjecture. The gravity dual of this superconformal field theory is given by type IIB superstring theory on

A finitely generated group is called Sasaki if it is the fundamental group of a compact Sasakian manifold, and a Kähler group is the fundamental group of a compact Kähler manifold.…

Let a Kähler group mean the fundamental group of a compact Kähler manifold, and let a compact Hodge manifold be a compact manifold whose fundamental group is under consideration as…