We compute a Bochner type formula for static three-manifolds and deduce some applications in the case of positive scalar curvature. We also explain in details the known general construction of the (Riemannian) Einstein $(n + 1)$-manifold associated to a maximal domain of a static $n$-manifold where the static potential is positive. There are examples where this construction inevitably produces an Einstein metric with conical singularities along a codimension-two submanifold. By proving versions of classical results for Einstein four-manifolds for the singular spaces thus obtained, we deduce some classification results for compact static three-manifolds with positive scalar curvature.

We show that on a Kähler manifold whether the $J$-flow converges or not is independent of the chosen background metric in its Kähler class. On toric manifolds we give a numerical characterization of when the $J$-flow converges, verifying a conjecture in [19] (M. Lejmi and G. Székelyhidi, “The $J$-flow and stability”) in this case. We also strengthen existing results on more general inverse $\sigma_k$ equations on Kähler manifolds.

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1 , n \geqslant 2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian on $(0, q)$-forms. We show that the spectral function of $\Box^{(q)}_{b}$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of $\Box^{(q)}_{b}$ in $] 0, \infty [$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szegő kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szegő kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR $S^1$ actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR $S^1$ action can be CR embedded into $\mathbb{C}^N$, for some $N \in \mathbb{N}$.

On compact manifolds of dimensions $4$ and more, we give a proof of Thurston’s existence theorem for foliations of codimension one; that is, they satisfy some $h$-principle in the sense of Gromov. Our proof is an explicit construction not using the Mather homology equivalence. Moreover, the produced foliations are minimal, that is, all leaves are dense. In particular, there exist minimal, $C^{\infty}$, codimension-one foliations on every closed connected manifold of dimension at least $4$ whose Euler characteristic is zero.