A Bernstein property for F relative parabolic affine hyperspheres
In the setting of relative affine differential geometry the author proves the following results: Theorem 1. Let \(u(\xi_1, \xi_2,\dots, \xi_n)\) be a strictly convex \(C^{\infty}\) function defined on \(\mathbb{R}^n\) satisfying the system of PDEs \[ \rho_{ij}=-{{F''}\over{F'}}\rho_i\rho_j+u^{kl}u_{ijk}\rho_l,\qquad 1\leq i,j\leq n. \] Denote the hypersurface by \[ M = \{(\xi,\xi_{n+1})|\xi_{n+1} = u(\xi), \xi\in \mathbb{R}^n\}. \] If the function \[ 2a^2-3(n + 2)a + 2a't +{{1}\over{4}}(n + 2)^2(5-n) \] with \(a = -{{F''(t)t}\over{F'(t)}}-1\), \(a'(t) = {\text{d a}\over\text{d t}}\) has a positive lower bound on \((0,+\infty)\), then \(M\) must be an elliptic paraboloid.
scientific article; zbMATH DE number 6965774
scientific article