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Maximum modulus sets in pseudoconvex boundaries. (English) Zbl 0772.32012
We quote the author’s abstract: “Let \(D\) be a strictly pseudoconvex domain in \(\mathbb{C}^ n\) with \(C^ \infty\) boundary. We denote by \(A^ \infty(D)\) the set of holomorphic functions in \(D\) that have a \(C^ \infty\) extension to \(\overline D\). A closed subset \(E\) of \(\partial D\) is locally a maximum modulus set of \(A^ \infty(D)\) if for every \(p\in E\) there exists a neighborhood \(U\) of \(p\) and \(f\in A^ \infty(D\cap U)\) such that \(| f|=1\) on \(E\cap U\) and \(| f|<1\) on \(\overline D\cap U\backslash E\). A submanifold \(M\) of \(\partial D\) is an interpolation manifold if \(T_ p(M)\subset T^ c_ p(\partial D)\) for every \(p\in M\), where \(T^ c_ p(\partial D)\) is the maximal complex subspace of the tangent space \(T_ p(\partial D)\). We prove that a local maximum modulus set for \(A^ \infty(D)\) is locally contained in totally real \(n\)-dimensional submanifolds of \(\partial D\) that admit a unique foliation by \((n-1)\)-dimensional interpolation submanifolds. Let \(D=D_ 1\times\cdots\times D_ r\subset\mathbb{C}^ n\) where \(D_ i\) is a strictly pseudoconvex domain with \(C^ \infty\) boundary in \(\mathbb{C}^{n_ i}\), \(i=1,\dots,r\). A submanifold \(M\) of \(\partial D_ 1\times\cdots\times\partial D_ r\) verifies the cone condition if \(\Pi_ p(T_ p(M))\cap\overline C[Jn_ 1(p),\dots,Jn_ r(p)]=\{0\}\) for every \(p\in M\), where \(n_ i(p)\) is the outer normal to \(D_ i\) at \(p\), \(J\) is the complex structure of \(\mathbb{C}^ n\), \(\overline C[Jn_ 1(p),\dots,Jn_ r(p)]\) is the closed positive cone of the real space \(V_ p\) generated by \(Jn_ 1(p),\dots,Jn_ r(p)\), and \(\Pi_ p\) is the orthogonal projection of \(T_ p(\partial D)\) on \(V_ p\). We prove that a closed subset \(E\) of \(\partial D_ 1\times\cdots\times\partial D_ r\) which is locally a maximum modulus set for \(A^ \infty(D)\) is locally contained in \(n\)-dimensional totally real submanifolds of \(\partial D_ 1\times\cdots\times\partial D_ r\) that admit a foliation by \((n-1)\)-dimensional submanifolds such that each leaf verifies the cone condition at every point of \(E\). A characterization of the local peak subsets of \(\partial D_ 1\times\cdots\times\partial D_ r\) is also given”.

MSC:
32A38 Algebras of holomorphic functions of several complex variables
32T99 Pseudoconvex domains
32V40 Real submanifolds in complex manifolds
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