On the Pick invariant, the affine mean curvature and the Gauss curvature of affine surfaces (Q1190279)

By departing from the so-called Affine Theorema Egregium for a nondegenerate unimodular affine surface \(M^ 2\subset\mathbb{R}^ 3\), which states that \({\mathfrak k}=H+J\), where \({\mathfrak k}\) is the Gauss curvature of the affine metric, \(H\) the affine mean curvature and \(J\) the Pick invariant, the authors study those affine surfaces where \({\mathfrak k}\), \(H\) and \(J\) are constant. They consider two separate cases regarding whether \(3{\mathfrak k}\neq H\) or \(3{\mathfrak k}=H\). Theorem. Let \(M^ 2\) be a nondegenerate surface in \(\mathbb{R}^ 3\) with constant mean curvature \(H\) and constant Pick invariant \(J\), further assumed to be \(>0\) by a suitable choice of orientation, then either \(3{\mathfrak k}=H<0\) or else \(M^ 2\) is a flat affine sphere. Next, the case where \({\mathfrak k}\) and \(H\) are constant is specifically considered, with \(3{\mathfrak k}=H\neq 0\). In this case a full classification separated in three cases is obtained: in one of them, given explicitly, it is shown that there is an open dense subset of \(M^ 2\) such that each connected component is affinely equivalent to an open part of the surface \((u,v,{1\over 2}(u^ 2+\varepsilon v^{-2/3}))\), \(\varepsilon=\pm 1\); the other two cases, not given explicitly, are characterized by using Radon's theorem. --- An alternative proof is also included for the case where \(M^ 2\) is locally strongly convex.