MAXIMUM MODULUS SET IN THE BOUNDARY OF RI GID DOMAINS IN C2

Let OMEGA = {(z, w) is-an-element-of C2, Re w + P (z) < 0}, where P is a real analytic function, subharmonic, with order 2 k at the origin a nd without harmonic terms in the lowest term of its Taylor development at the origin. Let E be a submanifold, totally real, dim E = 2, in th e boundary of OMEGA, the origin belonging to E. We prove the following results: when E is real analytic, we give a characterization of such E which are a A(omega)-maximum modulus set in a neighbourhood of the o rigin. When E is C- and verifies some geometric natural conditions, if E is a maximum modulus set for PSI in A(infinity) (Q) then psi can be extended holomorphically on a neighbourhood of the origin and E is re al analystic.