A Bernstein theorem for two-valued minimal graphs in dimension four

We prove a Bernstein-type theorem for two-valued minimal graphs in the four-dimensional Euclidean space $\mathbf{R}^4$. This states that two-valued functions defined on the entire $\mathbf{R}^3$, and whose graph is a minimal surface, must necessarily be linear. This is a two-valued analogue of the classical Bernstein theorem, which asserts that in dimensions up to $n+1 \leq 8$, an entire single-valued minimal graph is linear. The main contrast with the single-valued theory is the presence of a large set of singularities in the graphs of two-valued functions. Indeed two-valued minimal graphs are neither area-minimising, nor is the regularity theory of elliptic PDE directly available in this setting. We obtain structure results for the blowdown cones of two-valued minimal graphs, valid in dimension $n+1 \leq 7$, proving in particular that they are smoothly immersed away from an $(n-2)$-rectifiable set that includes its branch points. In dimension four we go further, and completely classify the possible blowdown cones using a combinatorial argument. We show that they must be a union of two three-dimensional planes: this is the key to the proof of the Bernstein theorem.