The affine complete hypersurfaces of constant Gauss-Kronecker curvature
Many higher order PDEs that arise from geometric problems can be solved in a natural way by geometric methods. In [\textit{A. Li, U. Simon} and \textit{B. Chen}, J. Reine Angew. Math. 487, 179--200 (1997; Zbl 0886.35065)] the authors established a two-step Monge-Ampère procedure for the solution of certain non-linear fourth order PDEs. The geometric background lies in Blaschke's hypersurface theory (see. e.g. chapter 1 in [\textit{A.-M. Li, U. Simon} and \textit{G. Zhao}, Global affine differential geometry of hypersurfaces. Berlin: Walter de Gruyter (1993; Zbl 0808.53002)]). In affine hypersurface theory there are different, non-equivalent notions of ``completeness'' that play a role for global investigations (for details see e.g. section 2.3 in [Zbl 0808.53002]). Consider a locally strongly convex hypersurface in affine space \(\mathbb R^{n+1}\), given by an immersion \(x: M \rightarrow \mathbb R^{n+1}\) of an oriented, connected \(C^{\infty}\)-manifold \(M\). Any additional Euclidean structure on \(\mathbb R^{n+1}\) induces a Euclidean first fundamental form \(g\) on \(M\). The completeness of this metric \(g\) does not depend on the special choice of the Euclidean structure of the ambient space, and this completeness is an affine notion; it is called Euclidean completeness of the affine hypersurface, while the completeness of the Blaschke metric is called affine completeness, and the completeness of the affine spherical metric, defined by the Weingarten form, is called Weingarten completeness. Within these non-equivalent notions, S. Y. Cheng and S. T. Yau (see section 2.3 in [Zbl 0808.53002]) were the first to state sufficient conditions that the Euclidean completeness implies the affine completeness; meanwhile one knows different and more general results of this type. The author of the present paper proves the following Main Theorem, extending Theorem 3 from [Zbl 0886.35065]: Given a locally strongly convex, hyperbolic and Weingarten complete hypersurface \(x\) with positive constant affine Gauss-Kronecker curvature; then {\parindent=8mm \begin{itemize}\item[(i)] \(x\) can be represented as the graph of some convex function \(f\); \item[(ii)] the Legendre domain \(\Omega\) of \(f\) is bounded and convex; \item[(iii)] the Legendre function \(u\) of \(f\) satisfies a certain Monge-Ampère equation of second order in~\(\Omega\); \item[(iv)] \(x\) is also Euclidean complete and affine complete. \end{itemize}} The proof is lengthy and difficult. Reviewer's remarks. While the results and methods of this paper are very interesting, the paper contains many misprints and grammatical mistakes; in such cases the editorial board of the journal should take the necessary steps for corrections before printing.
completeness in affine hypersurface theory
scientific article; zbMATH DE number 5601628