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Entire area-minimizing surfaces in R^4 are algebraic

Nick Edelen, Luis Atzin Franco Reyna, Paul Minter

TL;DR

The paper proves a sharp algebraic classification for entire 2-dimensional area-minimizing or stable surfaces in $\mathbb{R}^4$ with quadratic area growth, showing they are cut out by holomorphic polynomials whose total degree equals the density at infinity $\Theta$. It combines Rivière's uniqueness of tangent cones at infinity with Micallef's holomorphicity result to deduce holomorphicity after rigid motion and then applies Chow's theorem (GAGA) to obtain algebraicity; for area-minimizing $2$-currents with finite density, it gives a decomposition into a finite sum of algebraic currents with degrees $d_i$ and multiplicities $m_i$ satisfying $\sum_i m_i d_i=\Theta$, along with explicit bounds on the singular set and the genus of the regular part. The results link geometric measure theory with complex and algebraic geometry, yielding rigidity results and explicit genus/singularity bounds that depend only on $\Theta$, and provide concrete descriptions for low densities such as $\Theta\le 3$.

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.

Entire area-minimizing surfaces in R^4 are algebraic

TL;DR

The paper proves a sharp algebraic classification for entire 2-dimensional area-minimizing or stable surfaces in with quadratic area growth, showing they are cut out by holomorphic polynomials whose total degree equals the density at infinity . It combines Rivière's uniqueness of tangent cones at infinity with Micallef's holomorphicity result to deduce holomorphicity after rigid motion and then applies Chow's theorem (GAGA) to obtain algebraicity; for area-minimizing -currents with finite density, it gives a decomposition into a finite sum of algebraic currents with degrees and multiplicities satisfying , along with explicit bounds on the singular set and the genus of the regular part. The results link geometric measure theory with complex and algebraic geometry, yielding rigidity results and explicit genus/singularity bounds that depend only on , and provide concrete descriptions for low densities such as .

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.
Paper Structure (1 section, 5 theorems, 44 equations)

This paper contains 1 section, 5 theorems, 44 equations.

Table of Contents

  1. Proof of Main Result

Key Result

Theorem 1

Let $F:M^2\to \mathbb{R}^4\equiv \mathbb{C}^2$ be a connected orientable smooth stable minimal immersion without boundary, which: Then, we can write for some $A \in O(4)$, and $p:\mathbb{C}^2\to \mathbb{C}$ a holomorphic polynomial of degree $\Theta$. In particular, $F(M)$ is smooth and properly embedded away from a finite singular set, with bounds:

Theorems & Definitions (17)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Corollary 9
  • Theorem 1.1: Uniqueness of tangent cones at infinity, riviere
  • ...and 7 more