Isothermal Coordinate System
In the previous two sections, we have not been able to give a clear definition of minimal surfaces. In this section, we define minimal surfaces, compute their curvature, and find their Gauss map. Let us start with the definition.
Definition of Minimal Surface
We say a surface is a minimal surface if its mean curvature $H = 0$.
Definition of Isothermal Coordinates
Let $r = r(u, v)$ be a regular parametrized surface. We say that $r$ is isothermal if:
$$ E = G \quad \text{and} \quad F = 0, $$
where $E$, $F$, and $G$ are the coefficients of the first fundamental form.
Proposition: Conversion to Isothermal Coordinates
Let $r = r(u, v)$ be a regular parametrized surface. There exists a local change of coordinates $(u, v) \to (x, y)$ such that in the new coordinates the surface becomes isothermal. That is, the coefficients of the first fundamental form satisfy $E = G$ and $F = 0$.
Proof Sketch
Given a surface $r = r(u, v)$, the first fundamental form is:
$$ I = E , du^2 + 2F , du , dv + G , dv^2. $$
To achieve isothermal coordinates, we need:
$$ E , dx^2 + G , dy^2 \quad \text{and} \quad F = 0. $$
This can be done by solving the Beltrami equation for conformal mappings:
$$ \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} + \frac{\partial x}{\partial v} \frac{\partial y}{\partial u} = 0. $$
Details of the proof involve constructing such coordinates locally using complex analysis.
Minimal Surface Equation in Isothermal Coordinates
If $r = r(u, v)$ is an isothermal parametrization of a minimal surface, then $r$ satisfies the equation:
$$ r_{uu} + r_{vv} = 0. $$
This is equivalent to the condition that the parametrized surface is harmonic.
Gauss Map of a Minimal Surface
Let $r = r(u, v)$ be a minimal surface, and let $\mathbf{n}$ be the unit normal vector. The Gauss map $N : S \to \mathbb{S}^2$ sends each point $p$ on the surface to the corresponding unit normal vector $\mathbf{n}(p)$.
For minimal surfaces in isothermal coordinates, the Gauss map $N$ satisfies:
$$ N = \frac{r_u \times r_v}{| r_u \times r_v |}. $$
Example: Minimal Surfaces in Isothermal Coordinates
Catenoid: The catenoid can be parametrized in isothermal coordinates as:
$$ r(u, v) = (c \cosh(u) \cos(v), c \cosh(u) \sin(v), u). $$
Helicoid: The helicoid can be written as:
$$ r(u, v) = (u \cos(v), u \sin(v), c v). $$
Both examples demonstrate how isothermal coordinates simplify the representation of minimal surfaces.
Conclusion
Isothermal coordinates are a powerful tool for studying minimal surfaces. They simplify the equations governing these surfaces and provide a natural framework for exploring their geometric properties.
Weierstrass-Enneper Representation of Minimal Surface
With all of this background information about minimal surfaces, isothermal patches, harmonic functions, and holomorphic and meromorphic functions, the Weierstrass-Enneper representation for minimal surfaces may be constructed. Now we start from three complex functions.
For a given surface $r(u,v) = (x(u,v), y(u,v), z(u,v))$, we consider:
$$ \phi_1(w) = \frac{\partial x}{\partial u} - i \frac{\partial x}{\partial v}, \quad \phi_2(w) = \frac{\partial y}{\partial u} - i \frac{\partial y}{\partial v}, \quad \phi_3(w) = \frac{\partial z}{\partial u} - i \frac{\partial z}{\partial v}, $$
where $w = u + iv$. Now consider the sum of the square of the three complex functions, then we can get an interesting conclusion connected with the isothermal coordinate.
Theorem
For a given surface $r(u,v) = (x(u,v), y(u,v), z(u,v))$, we define:
$$ \phi_1(w) = \frac{\partial x}{\partial u} - i \frac{\partial x}{\partial v}, \quad \phi_2(w) = \frac{\partial y}{\partial u} - i \frac{\partial y}{\partial v}, \quad \phi_3(w) = \frac{\partial z}{\partial u} - i \frac{\partial z}{\partial v}. $$
The surface $r$ is an isothermal coordinate system if and only if $\phi_1(w)$, $\phi_2(w)$, $\phi_3(w)$ are holomorphic and satisfy:
$$ \phi_1^2(w) + \phi_2^2(w) + \phi_3^2(w) = 0. $$
When we study a minimal surface, the case where all of $\phi_1(w)$, $\phi_2(w)$, $\phi_3(w)$ are identically zero is excluded because it gives no surface. Hence we assume that $\phi_3$ is not identically zero. If we define:
$$ f = \phi_1 - i\phi_2, \quad g = \frac{\phi_3}{\phi_1 - i\phi_2}, $$
then $f$ is a holomorphic function and $g$ is a meromorphic function, since:
$$ \phi_1^2(w) + \phi_2^2(w) + \phi_3^2(w) = 0 \implies \phi_1 + i\phi_2 = -\frac{\phi_3^2}{\phi_1 - i\phi_2} = -fg^2. $$
Then we obtain:
$$ \phi_1 = \frac{1}{2}f(1 - g^2), \quad \phi_2 = \frac{i}{2}f(1 + g^2), \quad \phi_3 = fg. $$
Theorem
A simply connected minimal surface is expressed as the following:
$$ r(w) = \left( \text{Re} \int_0^w \frac{1}{2}f(1 - g^2) , dw, \text{Re} \int_0^w \frac{i}{2}f(1 + g^2) , dw, \text{Re} \int_0^w fg , dw \right), \quad w \in D, $$
where $D$ is the unit open disk:
$$ {w \in \mathbb{C} \mid |w| < 1} $$
or the total plane $\mathbb{C}$, $f(w)$ is a holomorphic function on $D$, and $g(w)$ is a meromorphic function on $D$.
Example (Enneper’s Surface)
Let $D = \mathbb{C}$, $w = u + iv$, $f(w) = 2$, $g(w) = w$. Then we have:
$$ x = \text{Re} \int_0^w (1 - w^2) , dw = \text{Re}(w - \frac{1}{3}w^3) = u + uv^2 - \frac{1}{3}u^3, $$
$$ y = \text{Re} \int_0^w i(1 + w^2) , dw = \text{Re}(i(w + \frac{1}{3}w^3)) = -v - u^2v + \frac{1}{3}v^3, $$
$$ z = \text{Re} \int_0^w 2w , dw = \text{Re}(w^2) = u^2 - v^2. $$