On polynomial convexity of compact subsets of totally-real submanifold in ${\mathbb{C}}^n$

Abstract

Let $K$ be a compact subset of a totally-real manifold $M$, where $M$ is either a ${\mathcal{C}}^2$-smooth graph in ${\mathbb{C}}^{2n}$ over ${\mathbb{C}}^n$, or $M=u^{-1}\{0\}$ for a ${\mathcal{C}}^2$-smooth submersion $u$ from ${\mathbb{C}}^n$ to ${\mathbb{R}}^{2n-k}$, $k\le n$. In this case we show that $K$ is polynomially convex if and only if for a fixed neighbourhood $U$, defined in terms of the defining functions of $M$, there exists a plurisubharmonic function $\Psi$ on ${\mathbb{C}}^n$ such that $K\subset \{\Psi <0\}\subset U$.