OAI.PinchedHartogs.metric_transfer

The theorem states (as an admitted result) that the proposition MetricTransfer holds. Work on the open unit ball in ℂ², with Hartogs domain hartogs(φ) = {(z,w) : |z|<1, exp(φ(z))|w|²<1} in ℂ²×ℂ for a real function φ on the ball. For all real constants 0<c≤C there is λ₀>0 such that for every φ that is C^∞ on the ball and satisfies ChartControl(φ,c,C) and ChartJets(φ,C), and every λ≥λ₀, the following hold. First, ChartControl says that after composing φ with each centered chart (a Möbius-type automorphism of the ball moving a radius-r point to the origin, r in [0,1), followed by any complex-linear isometry U), the complex Hessian of φ at the origin lies between c and C times the identity: c|v|² ≤ Hessian(v,v) ≤ C|v|². ChartJets says that in all these charts the first and mixed second derivatives (∂ and ∂̄ and ∂∂̄) of the Hessian entries at the origin have summed norm at most C. The conclusions are that hartogs(φ) is contractible; the metric given by the complex Hessian of the potential λ·(−log(1−|z|²)) − log(1 − exp(φ(z))|w|²) is a smooth Kähler metric on hartogs(φ), meaning it is smooth, Hermitian, positive definite, and satisfies the Kähler symmetry condition on first derivatives; this metric is geodesically complete; and there exist constants 0<A≤B such that all its real sectional curvatures on linearly independent pairs of tangent vectors lie in [−B,−A], so it is negatively pinched. The constants A and B may depend on φ and λ.