47 problems

Let $2\le n<N$, and let $F:\mathbb{B}^n\to\mathbb{B}^N$ be a proper holomorphic map admitting a $C^{N-n+1}$-smooth extension to the boundary. If the CR Gauss map of the boundary re…

For every real parameter $t>1$, does there exist a real-analytic CR-embedding $F_t:M_t^3\hookrightarrow\mathbb{C}^3$ such that $F_t(M_t^3)$ is a compact real-analytic strongly pseu…

Let $(M,\theta)$ be a closed pseudohermitian $(2n+1)$-manifold with $n\geq 2$. Assume, in addition, that the CR-Paneitz operator is nonnegative if $n=1$. Suppose there is a positiv…

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…

Fix a positive integer $n$. Let $(M,T^{1,0}M)$ be a closed strictly pseudoconvex CR manifold of dimension $2n+1$ admitting a pseudo-Einstein contact form. For each partition…

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 $M,M'\subset\mathbb{C}^n$, 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…

Local embeddability conjecture. A CR 3-space admits locally a closed, non-zero section of its canonical bundle if and only if it is locally embeddable.

Strong Ramadanov conjecture. If $\rho \in C^{n+2}(\overline{D})$, then $\partial D$ is locally spherical.

Let $M^3\hookrightarrow\mathbb{C}^3$ be an embedding, and suppose it has an isolated complex tangent. Isolated complex tangent conjecture. Every isolated complex tangent of the emb…

Let $(M,J,\theta)$ be a compact strictly pseudoconvex CR manifold of real dimension $2n+1$, and let $G$ be a compact subgroup of $I(M,\theta)$, the group of pseudo-Hermitian transf…

Wang's conjecture. If $1<\alpha\leqslant\frac{n+2}{n}$ and $\lambda\leqslant\frac{n}{2(\alpha-1)}$, then the only positive solution is $u\equiv\lambda^{\frac{1}{\alpha-1}}$. Otherw…

Let $(\widetilde{\mathcal{M}},\widetilde{\mathbf{c}},\widetilde{K})$ be an optical geometry of dimension $2m+2>4$ with twisting non-shearing congruence of null geodesics…

Let $H:\mathbb{S}^{2n-1}\rightarrow\mathbb{S}^{2N-1}$ be a sphere map with linearly independent components, and let $s>0$. Degree conjecture. If $H$ is $s$-degenerate, then $H$ has…

Let $M^{2m+1}$ be a closed pseudohermitian manifold with vanishing pseudohermitian torsion $A_{\alpha\beta}=0$ and Webster Ricci tensor satisfying … Suppose a positive function $f$…

Let $I$ be a natural pseudohermitian scalar invariant whose total integral is a secondary CR invariant. Here, a secondary CR invariant is a total integral invariant under changes o…

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 $(M,J,\theta)$ be a closed, spherical, strictly pseudoconvex CR $(2n+1)$-manifold with vanishing first Chern class … where $n\geq 2$. A contact form has CR-pluriharmonic $Q$-cu…

Let $M$ be a closed, strictly pseudoconvex CR $(2n+1)$-manifold with vanishing first Chern class … where $n\geq 2$. A pseudo-Einstein structure is a contact structure whose pseudoh…