Maximum modules sets of rigid domains of \(\mathbf C^ 2\) (Q1360060)

Let \(\Omega=\{(z,w) \in\mathbb{C}^2\), \(\text{Re} w+ P(z)<0\}\) where \(P\) is a real analytic function, subharmonic, with order \(2k\) at the origin and without harmonic term in the lowest term of its Taylor expansion at the origin. Let \(E\) be a totally real submanifold of dimension 2 in the boundary of \(\Omega\), containing the origin. When \(E\) is real analytic, we give a characterization of such \(E\) which are \(A^\omega\)-maximum modulus sets in a neighourhood of the origin. When \(E\) is \(C^\infty\) and verifies some natural geometric conditions, and if \(E\) is the maximum modulus set for \(\Psi\) in \(A^\infty (\Omega)\), then \(\Psi\) can be extended holomorphically on a neighbourhood of the origin and \(E\) is real analytic.