Locally strongly convex hypersurfaces with constant affine mean curvature
The authors study the Bernstein property of constant affine mean curvature surfaces in \(\mathbb R^3\). For any locally strongly convex surface given by a strictly convex function \(x_3= f(x_1,x_2)\) they consider the Riemannian metric defined by \(G^\#= \sum \frac{\partial^2f}{\partial x_i\partial x_j}\,dx_i\,dx_j\) on \(M=\{(x_1,x_2,f(x_1,x_2))\mid (x_1,x_2)\in \Omega\}\), where \(\Omega\) is a convex domain in \(\mathbb R^2\). Using a priori estimates on PDE they are able to show that if \(M\) is a constant affine mean curvature surface which is complete with respect to \(G^\#\), then \(M\) must be an elliptic paraboloid.
scientific article; zbMATH DE number 2165118