New examples of Calabi-Bernstein problems for some nonlinear equations.
A Robertson-Walker space is the product manifold \(M=I\times \mathbb{R}^2\) with the (warped) Lorentzian metric \[ ds^2=-dt^2+f(t)g, \] where \(t\in I\), \(f\) is a positive function in \(I\), and \(g\) is the usual metric of \(\mathbb{R}^2\). In the case \(f\equiv1\), we fall back in the usual Lorentz-Minkowski metric \(L^3\). It is assumed that \(M\) satisfies the null convergence condition (NCC), i.e., that \(\text{Ricc}(Z,Z)\geq 0\), for all vectors \(Z\) with \(\langle Z,Z\rangle=0\). This paper studies the Calabi-Bernstein problem for complete maximal surfaces in a RW-space satisfying NCC. One of its results is as follows: A Robertson-Walker space with fiber \(\mathbb{R}^2\) and not locally constant warping function \(f\) with \(\sup(f)<\infty\), which obeys NCC, admits at most one complete maximal surface which is a totally geodesic spacelike slice.
scientific article; zbMATH DE number 1959884