Constructing entire minimal graphs by evolving planes

We introduce an evolving-plane ansatz for the explicit construction of entire minimal graphs of dimension n ( \(n\ge 3\) $n\ge 3$ ) and codimension m ( \(m\ge 2\) $m\ge 2$ ). Under this ansatz, the minimal surface system reduces to the geodesic equation on the Grassmannian in affine coordinates. Geometrically, this equation dictates how the slope of an \((n-1)\) $(n-1)$ plane evolves as it sweeps out a minimal graph. This framework yields a large family of explicit entire minimal graphs of arbitrary dimension n and arbitrary codimension m. For each entire minimal graph, its conormal bundle gives rise to an entire special Lagrangian graph in \(\mathbb {C}^{n+m}\) ${\mathbb{C}}^{n+m}$ .