Two flat structures on minimal surfaces

TL;DR

This paper explains how two flat structures on minimal surfaces generate a harmonic function that uniquely characterizes Enneper's surface, providing insight into its geometric properties.

Contribution

It introduces a novel approach linking flat structures and harmonic functions to distinguish Enneper's surface among minimal surfaces.

Findings

Two flat structures induce a harmonic function on minimal surfaces.

The harmonic function uniquely characterizes Enneper's surface.

Provides a new perspective on the geometry of minimal surfaces.

Abstract

In this expository article, we illustrate how two independent flat structures on minimal surfaces induce a harmonic function, which captures the uniqueness of Enneper's surface.

Full text

Two flat structures on minimal surfaces

Hojoo Lee

momentmaplee@gmail.com

In this expository article, we illustrate how two flat structures on minimal surfaces induce a harmonic function, which captures the uniqueness of Enneper’s algebraic surface with total Gauss curvature $-4\pi$.

1. (1)

Let $\Sigma$ be a minimal surface in ${\mathbb{R}}^{3}$ with Gauss curvature $\mathcal{K}={\mathcal{K}}_{{\mathbf{g}}_{{}_{\Sigma}}}\leq 0$. Given the unit normal vector field $\mathbf{N}$ on $\Sigma$ and a constant unit vector field ${\mathbf{V}}(p)={\mathbf{V}}$ in ${\mathbb{R}}^{3}$, introduce the angle function

$\mathbf{N}_{\mathbf{V}}(p):={\langle\mathbf{N}(p),{\mathbf{V}}\rangle}_{{\mathbb{R}}^{3}},\quad p\in\Sigma.$ 2. (2)

Ricci’s flat structure. The do Carmo–Peng proof [4] of generalized Bernstein Theorem that planes are the only stable minimal surfaces in ${\mathbb{R}}^{3}$ uses the Ricci condition [4, Equation (2.9)]

$4\mathcal{K}={\triangle}_{{\mathbf{g}}_{{}_{\Sigma}}}\,\ln\left(-\mathcal{K}\right),$

which geometrically means Ricci’s Theorem [2, 5, 6] that the conformally changed metric

${\mathcal{G}}_{{\text{Ricci}}}={\left(-{\mathcal{K}}\right)}^{\frac{1}{2}}{\mathbf{g}_{{}_{\Sigma}}}$

is flat when $\mathcal{K}<0$. Ricci [1, p. 124] discovered that that every metric satisfying Ricci condition can be realized on a minimal surfaces in ${\mathbb{R}}^{3}$. See Lawson’s counterexample [5, Remark 12.1] for metics having a flat point, and also his higher codimensional generalization [6] of Ricci Theorem. 3. (3)

Chern’s flat structure. Chern’s non-complex-analytic proof [3] of Bernstein Theorem that planes are the only entire minimal graphs in ${\mathbb{R}}^{3}$ uses the intriguing equality (with $\mathbf{V}=(0,0,1)$):

$\mathcal{K}={\triangle}_{{\mathbf{g}}_{{}_{\Sigma}}}\,\ln\left(1+\mathbf{N}_{\mathbf{V}}\right),$

which geometrically means [3, Equation (3)] that the following conformally changed metric

${\mathcal{G}}_{{\text{Chern}}}={\left(1+\mathbf{N}_{\mathbf{V}}\right)}^{2}{\mathbf{g}}_{{}_{\Sigma}}$

is flat when $\mathbf{N}_{\mathbf{V}}>-1$. (In [3], $\mathbf{N}_{(0,0,1)}=\frac{1}{\,\sqrt{\,1+{f_{x}}^{2}+{f_{y}}^{2}\,}\,}>0$ on the minimal graph $z=f(x,y)$.) 4. (4)

Induced harmonic function. When ${\mathcal{K}}<0$ and $\mathbf{N}_{\mathbf{V}}>-1$, since metrics $\widetilde{g}={\mathcal{G}}_{\text{Ricci}}$ and ${\mathcal{G}}_{\text{Chern}}=\star\widetilde{g}=\frac{{\left(1+{\mathcal{N}}_{\mathbf{V}}\right)}^{2}}{{\left(-\mathcal{K}\right)}^{\frac{1}{2}}}{\mathcal{G}}_{\text{Ricci}}$ are flat, by the curvature formula of conformally changed metric

$0={\mathcal{K}}_{\star\widetilde{g}}=\frac{1}{\star}{\mathcal{K}}_{\widetilde{g}}-\frac{1}{2\star}{\triangle}_{\widetilde{g}}\ln\star=0-\frac{1}{2\star}{\triangle}_{\widetilde{g}}\ln\star,$

we find the harmonicity (with respect to metrics ${\mathcal{G}}_{\text{Ricci}}$, ${\mathcal{G}}_{\text{Chern}}$, ${\mathbf{g}_{{}_{\Sigma}}}$) of our Chern-Ricci function

$\ln\star=\ln\left(\frac{{\left(1+{\mathcal{N}}_{\mathbf{V}}\right)}^{2}}{{\left(-\mathcal{K}\right)}^{\frac{1}{2}}}\right).$ 5. (5)

Chern, Ricci, and Enneper [7, Theorem 3.1]. If a minimal surface in ${\mathbb{R}}^{3}$ has a constant Chern-Ricci function, then it is (a part of) an Enneper surface, up to isometries and homotheties.

Enneper’s surfaces have no flat points. We list more uniqueness results. Bernstein-Mettler Theorem [2] characterizes the Enneper surface as the nonflat minimal surface with vanishing entropy differential. Pérez [8] gave a novel variational characterization of half of the Enneper surface containing a straight line. White’s original remarkable conjecture [10] states that the half Enneper surface is the unique properly embedded nonflat orientable area minimizing surface with line boundary and quadratic area growth.