Global plurisubharmonic defining functions (Q5954536)

Let \(D\subset\mathbb C^n\) be a bounded pseudoconvex domain with real analytic boundary. Assume that \(D\) is linearly regular, i.e. there is no non-trivial smooth curve \(\gamma\) in \(\partial D\) such that \(L_{\gamma(t)}(r,\gamma'(t))=0\) for all \(t\), where \(r\) is a local defining function and \(L_p(r,t):=\sum_{j,k=1}^n \frac{\partial^2r}{\partial z_j\partial\overline z_k}(p)t_j\overline t_k\) denotes the Levi form. NEWLINENEWLINENEWLINEThe main result of the paper says that if for each \(a\in\partial D\) there exists a neighborhood \(U_a\) of \(a\) and a local smooth defining function \(r_a\) with \(L_p(r_a,t)\geq 0\), \(p\in\partial D\cap U_a\), \(t\in\mathbb C^n\), then there exists a global smooth defining function \(r\) with \(L_p(r,t)\geq 0\), \(p\in\partial D\), \(t\in\mathbb C^n\).