The topology of rationally and polynomially convex domains (Q2514522)

For a compact set \(K\subset\mathbb{C}^n\), \(n\geq3\), its polynomial hull is defined as \[ \widehat{K}_{\mathcal{P}}:=\Big\{z\in\mathbb{C}^n\Big|\;|P(z)|\leq\max\limits_{u\in K}|P(u)|\text{ for all complex polynomials }P\text{ on }\mathbb{C}^n\Big\}, \] and its rational hull as \[ \widehat{K}_{\mathcal{R}}:=\Big\{z\in\mathbb{C}^n\Big|\;|R(z)|\leq\max\limits_{u\in K}|R(u)|\text{ for all rational functions }R=\frac{P}{Q},\;Q\big|_K\not=0\Big\} \] In this paper, the authors give necessary and sufficient conditions on the topology of \(K\) with smooth boundary to be isotopic to a rationally or polynomially convex domain. Theorem 1.2. A compact domain \(K\subset\mathbb{C}^n\), \(n\geq3\), is smoothly isotopic to a rationally convex domain if and only if it admits a defining Morse function without critical points of index \(> n\). A similar theorem is proved for polynomially convex domains. These results are consequences of a more precise theorem for flexible Stein domains.