Narrow Results

We prove the compactness of the set of solutions to the CR Yamabe problem on a compact strictly pseudoconvex CR manifold of dimension three whose blow-up manifolds at every point have positive p-mass. As a corollary, we deduce that compactness holds for CR-embeddable manifolds which are not CR-equivalent to $S^3$. The theorem is proved by blow-up analysis.

We show that a compact orientable 4-manifold M has a CR regular immersion into ℂ³ if and only if both its first Pontryagin class p₁(M) and its Euler characteristic χ(M) vanish, and has a CR regular embedding into ℂ³ if and only if in addition the second Stiefel–Whitney class w₂(M) vanishes.

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1$, $n\ge 1$ with a transversal CR $S^1$-action on $X$. We introduce the Fourier components of the Ray–Singer analytic torsion on $X$ with respect to the $S^1$-action. We establish an asymptotic formula for the Fourier components of the analytic torsion with respect to the $S^1$-action. This generalizes the asymptotic formula of Bismut and Vasserot on the holomorphic Ray–Singer torsion associated with high powers of a positive line bundle to strongly pseudoconvex CR manifolds with a transversal CR $S^1$-action.

Let $X$ be a compact connected orientable Cauchy–Riemann (CR) manifold with the action of a compact Lie group $G$. Under natural pseudoconvexity assumptions we show that the CR Guillemin–Sternberg map is an isomorphism at the level of Sobolev spaces of CR functions, modulo a finite-dimensional subspace. As application we study this map for holomorphic line bundles which are positive near the inverse image of $0$ by the momentum map. We also show that “quantization commutes with reduction” for Sasakian manifolds.