LedD be a strictly pseudoconvex domain in ℂn withC∞ boundary. We denote byA∞(D) the set of holomorphic functions inD that have aC∞ extension to\(\bar D\). A closed subsetE of ∂D is locally a maximum modulus set forA∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A∞(D∩U) such that |f|=1 onE∩U and |f|<1 on\(\bar D \cap U\backslash E\). A submanifoldM of ∂D is an interpolation manifold ifTp(M)⊂Tpc(∂D) for everyp∈M, whereTpc(∂D) is the maximal complex subspace of the tangent spaceTp(∂D). We prove that a local maximum modulus set forA∞(D) is locally contained in totally realn-dimensional submanifolds of ∂D that admit a unique foliation by (n−1)-dimensional interpolation submanifolds. LetD =D1 x ... xDr ⊂ ℂn whereDi is a strictly pseudoconvex domain withC∞ boundary in ℂni,i=1,…,r. A submanifoldM of ∂D1×…×∂Dr verifies the cone condition if\(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, whereni(p) is the outer normal toDi atp, J is the complex structure of ℂn,\(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceVp generated byJn1(p),…,Jnr(p), and IIp is the orthogonal projection ofTp(∂D) onVp. We prove that a closed subsetE of ∂D1×…×∂Dr which is locally a maximum modulus set forA∞(D) is locally contained inn-dimensional totally real submanifolds of ∂D1×…×∂Dr that admit a foliation by (n−1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ∂D1×…×∂Dr is also given.