Abstract
Let M be a real hypersurface of class C2 in Cn, n ≥ 2, and let δM(z) be the Euclidean distance from z ∈ Cn to M. In this paper we give an explicit representation in complex tangential direction for the Levi form of the function δM (or -log δM) by Hermitian and symmetric matrices determined by a local defining function of M. As its application, we also show that, if M is defined by a C2-function &rho with d&rho ≠ 0, then the function -log δM is strictly plurisubharmonic in complex tangential direction near M if and only if M is Levi-flat and the symmetric matrix (δ2&rho/δzi δzj ) of degree n has maximal rank n - 1 on the complex tangent subspace T z1,0 (M) ⊂ T z1,0 (Cn) for each z ∈ M as a linear map. Moreover, we can get directly the well-known Levi condition as the condition in order that -log δM is weakly plurisubharmonic in complex tangential direction near and in one side of M.
