OAI.PinchedHartogs.main_theorem

The theorem states that there exist a subset M of ℂ³ (realized as ℂ² × ℂ) and a matrix-valued function g assigning a 3×3 complex matrix to each point, together with real numbers A and B, such that M is open, nonempty and contractible, and g is a Kähler metric on M, meaning g is C^∞ on M, Hermitian at each point (g_ij = conj(g_ji)), positive definite (the Hermitian form of g at u has positive real part for every nonzero u), and satisfies the closedness condition ∂g_kj/∂z_i = ∂g_ij/∂z_k, with ∂ the Wirtinger derivative (1/2)(∂_x − i∂_y). Moreover the metric is geodesically complete on M: for every point p in M and every vector v there is a C^∞ curve γ: ℝ → M with γ(0)=p, γ'(0)=v that satisfies the geodesic equation γ''k + Σ Γ^k{ac} γ'_a γ'_c = 0 for all t, where the Christoffel symbols are built from the inverse of g and the ∂ derivatives of g. The metric is also negatively pinched: 0 < A ≤ B and, for every point of M and every real-linearly independent pair u, v, the sectional curvature, computed from the explicit curvature tensor formula in the source, lies between −B and −A. In addition, M has no bounded coordinates: there is no map F from ℂ³ to ℂ³, holomorphic on a neighbourhood of every point of M and bounded in every component on M, whose complex Jacobian determinant is nonzero at every point of M. Finally, M is not biholomorphic to a bounded domain, that is, there is no bounded open set N with holomorphic maps F on M and G on N, each analytic near the respective set, mapping M into N and N into M, which are mutually inverse.