112 problems

Let $M$ be a CR manifold with a pseudo-Einstein contact form, and let a pseudo-hermitian invariant be defined for such contact forms. Suppose that its integral gives a global secon…

Zaitsev's conjecture. 1-jets suffice to determine all local automorphisms of $M$, except when $M$ is biholomorphic to the sphere at a generic point, which is the case of $S_k$.

Let $M\subset P_n(\mathbb{C})$ be a smooth hypersurface, where $n\geq 2$. Global minimality conjecture. Every such $M$ is globally minimal. This conjecture would remove the global…

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 $M\subset\mathbb C^{n+k}$ be a totally nondegenerate CR model of length $\rho\geq 3$, with graded infinitesimal CR automorphism algebra … Here, rigidity means that the positive…

Let $(M^3,J,\theta)$ be a closed three-dimensional pseudohermitian manifold, and let $[\theta]$ denote its contact class. A contact form in this class is said to have vanishing CR…

Let $Y$ be a contact manifold with embeddable CR structures $J_1$ and $J_2$, and let $X_{J_1}$ and $X_{J_2}$ be their nonsingular Stein fillings. Glue these fillings along their co…

Properness conjecture. The action of $\scr A(M)$ on $M$ should be proper unless $M$ is CR equivalent to the sphere or to the Heisenberg group with its standard CR structure.

Let $M\subset\mathbb{C}^n$ be a generic submanifold consisting of a single global CR orbit. Jöricke–Tumanov wedge-extension conjecture. Extension of CR functions to a wedge should…

Let $M\subset \mathbb{C}^{n+1}$ be a connected real-analytic holomorphically nondegenerate hypersurface, with $n\geq 1$. For each point $p\in M$, let a germ at $p$ of a local real-…

CR Obata conjecture. These assumptions should imply that $M$ is CR isomorphic to $S^3$. This is the equality case of the lower eigenvalue estimate proved in the paper, analogous to…

Let $M$ be a strictly pseudoconvex CR manifold with contact form $\theta$, Tanaka–Webster connection $\nabla$, and associated metric $g_\theta$. Fix $x\in M$ and a $2$-plane…

Let $(Y,H)$ be a compact, 3-dimensional contact manifold, and let $\mathcal{S}_0$ and $\mathcal{S}_1$ be Szegő projectors associated with embeddable, strictly pseudoconvex CR-struc…

Let $M$ be a compact strictly pseudoconvex Cauchy-Riemann $3$-manifold and let $\theta$ be a contact form on $M$. Write $R$ and $\tau$ for the Tanaka-Webster curvature and torsion…

Let $D_1,D_2\subset\mathbb{C}^n$ be smoothly bounded domains, and let a biholomorphism between them induce a map between their boundaries. Fefferman's boundary regularity conjectur…

Let $M,M'\subset\mathbb{C}^n$, with $n\geq 2$, be globally minimal real analytic hypersurfaces. Let $h:M\to M'$ be a continuous CR mapping whose holomorphic extension to a global o…

Let $N$ and $N'$ be integers with $2\le N\le N'$. Let $M\subset\mathbb C^N$ and $M'\subset\mathbb C^{N'}$ be real-analytic strictly pseudoconvex hypersurfaces, and let a formal emb…

Let $M\subset\mathbb C^N$ be a real analytic generic submanifold and let $p\in M$ be a point of finite type. Let $M'\subset\mathbb C^{N'}$ be a real analytic set containing no nont…

Let $M$ and $M'$ be connected real analytic hypersurfaces through $0$ in $\mathbb C^2$, and suppose that $M$ is of finite type at some point. Let…

Let $M\subset\mathbb C^N$ be a real analytic CR submanifold. Let $p\in M$, let $M'\subset\mathbb C^N$ be a real analytic submanifold with $\dim_{\mathbb R}M=\dim_{\mathbb R}M'$, an…

Let $M\subset\mathbb C^N$ be a real analytic CR submanifold, let $p\in M$ be a point of finite type, and let $K$ be a positive integer. Let $M'\subset\mathbb C^{N'}$ be a real anal…

Baouendi–Rothschild–Zaitsev approximation conjecture. The holomorphic mapping $H$ with this approximation property should exist under these hypotheses. This would generalize the kn…

Let $M$ be a strictly pseudoconvex CR manifold of dimension $2N-1$, let $\theta$ be a contact form, and let $S$ be a scalar pseudohermitian invariant satisfying the CR pluriharmoni…

Let $M\subset \mathbb{C}^N$ and $M'\subset \mathbb{C}^{N'}$ be real-analytic strongly pseudoconvex hypersurfaces, with $N'>N$. A formal embedding is a formal holomorphic embedding…

Tightness conjecture. Every spherical structure on a closed $3$-manifold is tight.