proceedings of the american mathematical society Volume %, Number 3. March 1986 THE MAXIMUM MODULUS PRINCIPLE FOR CR FUNCTIONS ANDREI IORDAN Abstract. Let M be a CR submanifold of C" without extreme points. Then, the modulus of any CR function on M cannot have a strong local maximum at any point of M. Preliminaries. Let M be a smooth manifold embedded as a locally closed real submanifold of C". We denote by dM the tangential Cauchy-Riemann operator on M induced by the Cauchy-Riemann operator 3 on C" and with HT (AÍ) the holomorphic tangent space to M at a point p g M. Let us recall some of the definitions and results of [4]. Definition 1. dM obeys the local maximum modulus principle on M if given any open connected set U in M and any u differentiable in U such that dMu = 0 on U, then u cannot have a (weak) local maximum at any point of U unless u is constant ont/. Definition 2. We call a point p g M an extreme point of M if there exists a local holomorphic coordinate system z = (z,,..., zn) in a neighborhood U of p such that z(p) = 0 and M n U C {z\yl > 0}. Here we assume that locally near p, M is not contained in any CA for k < n. Definition 3. (i) For any p g M and X g HT (M) set Z = X - iY, where Y = JX e HT^(M) and J is the multiplication with (-1)1/2 which defines the complex structure on R2". The Lev i form at p assigns to Z the normal vector L (Z) defined by L (Z) = Bp(X, X) + Bp(Y, Y), where Bp is the second fundamental form of M at p. (ii) We denote N (M) as the normal space of M at p. For any | G Np(M) the map Lp defined by L\{Z) = (Lp(Z), £> is called rft<? Levi form of M at p in the £ direction. Here ( , ) represents the real inner product in R2". We assume that p = 0 and codimRM = q. Then in a neighborhood £/ of the origin there are smooth real functions px,...,pq such that dpY A • • • Adpq^0 =£0 and Mn/7= (zG f/|Pl(z)= ••■ =P(/(z) = 0}. We may assume that i/p,(0),..., dpq(Q) are orthonormal. Received by the editors September 27, 1984 and, in revised form, March 13, 1985. 1980 Mathematics Subject Classification. Primary 32F25. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 465 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use