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 $\mathbb{R}^{2n-k}$, $k\leq n$. 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 $K\subset \{\Psi<0\}\subset U$.