Entire area-minimizing surfaces in R^4 are algebraic

TL;DR

This paper proves that all entire 2D area-minimizing surfaces in four-dimensional space are algebraic, characterized by holomorphic polynomials, and relates their geometric complexity to their density at infinity.

Contribution

It classifies entire area-minimizing surfaces in R^4 as algebraic, linking their structure to holomorphic polynomials and density at infinity.

Findings

Surfaces are algebraic and cut out by holomorphic polynomials.

Bounds on singular set size and genus are established.

Classification based on density at infinity.

Abstract

We classify entire 2-dimensional area-minimizing or stable surfaces in R^4 with quadratic area growth as algebraic, cut out by a finite union of holomorphic polynomials whose collective degrees are controlled by the density at infinity. As a consequence, we obtain bounds on the singular set size and genus in terms of the density at infinity.