Constructing entire minimal graphs by evolving planes

We introduce an evolving-plane ansatz for the explicit construction of entire minimal graphs of dimension $n$ ($n\geq 3$) and codimension $m$ ($m\geq 2$), for any odd integer $n$. 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 $(n-1)$ plane evolves as it sweeps out a minimal graph. This framework yields a rich family of explicit entire minimal graphs of odd dimension $n$ and arbitrary codimension $m$. For each entire minimal graph, its conormal bundle gives rise to an entire special Lagrangian graph in $\mathbb{C}^{n+m}$.

💡 Research Summary

This paper introduces a novel and powerful method, termed the “evolving-plane ansatz,” for the explicit construction of entire minimal graphs—minimal submanifolds defined over the entire Euclidean space R^n—of odd dimension n (n ≥ 3) and arbitrary codimension m (m ≥ 2).

The central idea is to represent the graph of a vector-valued function f: R^n → R^m in a specific linear form: f^α(x1,…, x_{n-1}, t) = Σ_{i=1}^{n-1} x_i * z_i^α(t), where α=1,…,m. Here, t is treated as a time parameter, and the “slope matrix” Z(t) =