Convex domains with locally Levi-flat boundaries

Abstract

It is shown that a domain in $\C^n$ which is locally convex and has ${\mathcal{C}}^1$-smooth Levi-flat boundary is locally linearly equivalent to a Cartesian product of a planar domain and $\C^{n-1}.$ This result does not extend to the case where the Levi form has locally constant rank for some $k<n-1$.

We learned that the main result of the paper is a special case of the result by V. K. Beloshapka and S. N. Bychkov in "A property of convex hypersurfaces in C^n", Mat. Zametki 40 (1986), no. 5. Interested readers should refer to that work