On a Domain in C with Generic Piecewise Smooth Levi-flat Boundary and Non-compact Automorphism Group

The boundary bD of a bounded domain D in C is called piecewise smooth if there exists a neighborhood U of D and ρk ∈ C (U), 1 ≤ k ≤ m, such that D = {q ∈ U ; ρk(q) < 0, 1 ≤ k ≤ m} and dρk1∧. . .∧dρkl 6= 0 on ∩ l j=1Skj for any distinct k1, . . . , kl ∈ {1, . . . , m}, where Sj = {q ∈ U ; ρj(q) = 0}. It is called generic piecewise smooth if ∂ρk1∧. . .∧∂ρkl 6= 0 on ∩j=1Skj . The boundary bD is called (generic) piecewise smooth Levi-flat if each Sj is in additional Levi-flat (See Section 3). We will call {ρj ; 1 ≤ j ≤ m} a defining system of D and each Sj a defining hypersurface of D. When D is convex, the above result was obtained by K.-T. Kim [Kim1] (see [W2] for related results). Kim’s proof uses a refine version of the rescaling method introduced by Frankel [Fra]. It was proved by Pinchuk [P] that a homogeneous bounded domain with piecewise smooth boundary is biholomorphic to a product of balls. Note that the noncompact condition in the above theorem is weaker than the homogeneous condition in Pinchuk’s result. In the latter case, one can choose a special boundary accumulation point that has properties similar to those possessed by a strictly pseudoconvex boundary point. See [Kod1, 2] and [CS] for results along this line. We remark that the simply-connected condition on D cannot be dropped. For example, the product of a disc and an annulus has generic piecewise smooth Levi-flat boundary and non-compact automorphism group. However, it is not biholomorphic to the bidisc. For motivation and background on the subject, we refer readers to [W1,2], [R], [GK1-3], [BP1,2], [Kim1,2], [FIK], and references therein.