Bernstein theorems for complete \(\alpha\)-relative extremal hypersurfaces (Q1941758)

The following main theorem is proved. { Theorem.} Let \(y: M\to \mathbb R^{n+1}\) be a locally strongly convex \(\alpha\) relative extremal hypersurface, complete with respect to the \(\alpha\)-metric \(G^{(\alpha)}\), which is given by a locally strongly convex function: \(x_{n+1}=f(x_1,\cdots,x_n)\). Then there is a positive constant \(K(n)\) depending only on the dimension \(n\), such that \(|\alpha|>K(n)\) implies that \(M\) is an elliptic paraboloid.