Suppose $\varphi$ is a real analytic plurisubharmonic exhaustion function on a connected noncompact complex manifold $X$. The main result is that if the real analytic set of points at which $\varphi$ is not strongly $q$-convex is of dimension at most $2q+1$, then almost every sufficiently large sublevel of $\varphi$ is strongly $q$-convex as a complex manifold. For $X$ of dimension $2$, this is a special case of a theorem of Diederich and Ohsawa. A version for $\varphi$ real analytic with corners is also obtained.

Let $D\subset \subset {ℂ}^n,n\ge 2$, be a domain with $C^2$-boundary and $K\subset \partial D$ be a compact set such that $\partial D\setminus K$ is connected. We study univalent analytic extension of CR-functions from $\partial D\setminus K$ to parts of $D$. Call $K$ CR-convex if its $A\left(D\right)$-convex hull, $A\left(D\right)-\mathrm{hull}\left(K\right)$, satisfies $K=\partial D\cap A\left(D\right)-\mathrm{hull}\left(K\right)$ ($A\left(D\right)$ denoting the space of functions, which are holomorphic on $D$ and continuous up to $\partial D$). The main theorem of the paper gives analytic extension to $\partial D\setminus A\left(D\right)-\mathrm{hull}\left(K\right)$, if $K$ is CR- convex.

We consider a singularity perturbed nonlinear differential equation $\epsilon u^{\text{'}}=f\left(x\right)u++\epsilon P(x,u,\epsilon )$ which we suppose real analytic for $x$ near some interval $[a,b]$ and small $\left|u\right|$, $\left|\epsilon \right|$. We furthermore suppose that 0 is a turning point, namely that $xf\left(x\right)$ is positive if $x\ne 0$. We prove that the existence of nicely behaved (as $ϵ\to 0$) local (at $x=0$) or global, real analytic or $C^{\infty }$ solutions is equivalent to the existence of a formal series solution $\sum u_n\left(x\right){\epsilon }^n$ with $u_n$ analytic at $x=0$. The main tool of a proof is a new “principle of analytic continuation” for...

Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f:W:=(A\times Y)\cup (X\times B)\to Z$ extends to a holomorphic mapping $\widehat{f}$ on $\widehat{W}:=\left\{(z,w)\in X\times Y:\tilde{\omega }(z,A,X)+\tilde{\omega }(w,B,Y)\<1\right\}$ such that $\widehat{f}=f$ on $W\cap \widehat{W},$ where $\tilde{\omega }(·,A,X)$ (resp. $\tilde{\omega }(·,B,Y))$ is the plurisubharmonic measure of $A$ (resp. $B$) relative to $X$ (resp. $Y$). Generalizations...

Let M be a generic CR submanifold in ${ℂ}^{m+n}$, m = CR dim M ≥ 1, n = codim M ≥ 1, d = dim M = 2m + n. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f{,}_f,\left[{\Gamma }_f\right])$, where: 1) $f{:}_f\to Y$ is a ¹-smooth mapping defined over a dense open subset ${}_f$ of M with values in a projective manifold Y; 2) the closure ${\Gamma }_f$ of its graph in ${ℂ}^{m+n}\times Y$ defines an oriented scarred ¹-smooth CR manifold of CR dimension m (i.e. CR outside a closed thin set) and 3) $d\left[{\Gamma }_f\right]=0$ in the sense of currents. We prove that $(f{,}_f,\left[{\Gamma }_f\right])$ extends meromorphically...