Entire area-minimizing surfaces in ${\mathbf{R}}^n$ of density two are quadratic or planar

Abstract

We classify any entire area-minimizing surface $M^2\subset {\mathbf{R}}^n$ with density $2$ at infinity as either planar or algebraic, namely the zero set of a quadratic holomorphic polynomial (up to rigid motion) and contained within an affine copy of ${\mathbf{R}}^4$. In particular, $M$ is either planar, or smooth, multiplicity one, and genus zero.

44 pages