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

TL;DR

This paper characterizes polynomial convexity of compact subsets within certain totally-real submanifolds in complex space using the existence of specific plurisubharmonic functions.

Contribution

It provides a new criterion for polynomial convexity of compact sets in totally-real manifolds based on plurisubharmonic functions.

Findings

Polynomial convexity is equivalent to the existence of a suitable plurisubharmonic function.

The criterion applies to manifolds that are either smooth graphs or level sets of smooth submersions.

The results extend understanding of polynomial convexity in complex analysis and geometry.

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 $K\subset \{\Psi<0\}\subset U$.