The space of parabolic affine spheres with fixed compact boundary (Q1568085)

It is well-known that the graph of every solution \(f\) of the equation of Monge-Ampère type \(\text{det}({\partial^2f\over \partial x_i\partial x_j})= 1\) is an improper affine sphere. The authors investigate the moduli space \({\mathcal M}\) of all solutions of this equation on the exterior \(\Omega\) of a plane Jordan curve of class \({\mathcal C}^\infty\) which fulfill the boundary condition \(f|_{\partial\Omega}= \varphi\in{\mathcal C}^\infty(\partial\Omega)\) and are themselves of class \({\mathcal C}^{2,\alpha}(\overline\Omega)\). It turns out that \({\mathcal M}\) (if not empty) is a 5-dimensional differentiable manifold. In the proof, the main tool is the Lewy transformation \(L_f: z= L_f(w):= w+2{\partial f\over\partial\overline w}\) together with the (holomorphic) Lewy function \(F:F(z):= \overline w- 2{\partial f\over\partial w}\) (\(w:= x_1+ ix_2\)).