Constructing entire minimal graphs by evolving planes

TL;DR

This paper presents a new method for explicitly constructing entire minimal graphs of odd dimension and arbitrary codimension by evolving planes, linking minimal surface systems to geodesic equations on Grassmannians.

Contribution

Introduces an evolving-plane ansatz that simplifies the minimal surface system to a geodesic equation, enabling explicit construction of a broad family of minimal graphs.

Findings

Provides explicit examples of entire minimal graphs in higher dimensions and codimensions.

Establishes a connection between minimal graphs and special Lagrangian geometry.

Enables systematic generation of minimal graphs via geodesic equations on Grassmannians.

Abstract

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 $\mathbb{C}^{n+m}$.