Weierstrass representation of a minimal surface

Let $M$ be a Riemann surface. A harmonic conformal mapping $X : M \rightarrow {\bf R} ^ { n }$ then defines a minimal surface in ${\bf R} ^ { n }$, $n \geq 3$ (cf. also Harmonic function; Conformal mapping). Let $z = u + i v$ be local isothermal coordinates; then

\begin{equation*} \sum _ { j = 1 } ^ { n } \Bigl( \frac { \partial X _ { j } } { \partial z } \Bigr) ^ { 2 } = 0. \end{equation*}

Since $X$ is harmonic,

\begin{equation*} \omega _ { j } = 2 \frac { \partial X _ { j } } { \partial z } d z \end{equation*}

is a holomorphic $1$-form on $M$. Hence any (branched) minimal surface in ${\bf R} ^ { n }$ can be given by $n$ meromorphic $1$-forms $\omega _ { j }$ satisfying $\sum _ { j = 1 } ^ { n } \omega _ { j } ^ { 2 } = 0$, and $X$ can be expressed as

\begin{equation} \tag{a1} X ( p ) = \operatorname { Re } \int _ { p _ { 0 } } ^ { p } ( \omega _ { 1 } , \ldots , \omega _ { n } ). \end{equation}

Such an $X$ is well defined on $M$ if and only if for any loop $C$ in $M$,

\begin{equation} \tag{a2} \operatorname { Re } \int _ { C } ( \omega _ { 1 } , \dots , \omega _ { n } ) = ( 0 , \dots , 0 ). \end{equation}

For $n = 3$, one gets a meromorphic function $g$ and a meromorphic $1$-form $ \eta $,

\begin{equation*} g = - \frac { \omega _ { 1 } + i \omega _ { 2 } } { \omega _ { 3 } } = \frac { \omega _ { 3 } } { \omega _ { 1 } - i \omega _ { 2 } } , \eta = g ^ { - 1 } \omega _ { 3 }. \end{equation*}

On the other hand, given a meromorphic function $g$ and a meromorphic $1$-form $ \eta $ on $M$, define

\begin{equation} \tag{a3} \omega _ { 1 } = \frac { 1 } { 2 } ( 1 - g ^ { 2 } ) \eta , \omega _ { 2 } = \frac { i } { 2 } ( 1 + g ^ { 2 } ) \eta , \omega _ { 3 } = g \eta ; \end{equation}

then $\sum _ { j = 1 } ^ { 3 } \omega _ { j } ^ { 2 } = 0$. Thus, (a3) together with (a1) defines a minimal surface in $\mathbf{R} ^ { 3 }$ and is called the Weierstrass representation of the minimal surface via the Weierstrass data $( g , \eta )$.

The meromorphic function $g$ has the geometric meaning that it is the composite of the spherical mapping (or unit normal vector) $N : M \rightarrow S ^ { 2 }$ and the stereographic projection from the north pole, where

\begin{equation*} N = \frac { 1 } { | g | ^ { 2 } + 1 } ( 2 \operatorname { Re } g , 2 \operatorname { Im } g , | g | ^ { 2 } - 1 ) \end{equation*}

and $g$ is also called the Gauss map of the minimal surface.

The first fundamental form and the Gaussian curvature of the surface $X ( M )$ can be expressed via $( g , \eta )$,

\begin{equation*} d s ^ { 2 } = \frac { 1 } { 4 } ( 1 + | g | ^ { 2 } ) ^ { 2 } | \eta | ^ { 2 } = \frac { 1 } { 2 } \sum _ { j = 1 } ^ { 3 } | \omega _ { j } | ^ { 2 }, \end{equation*}

\begin{equation*} K = - \left( \frac { 4 | d g | } { ( 1 + | g | ^ { 2 } ) ^ { 2 } | \eta | } \right) ^ { 2 }. \end{equation*}

Hence $X ( M )$ is a regular surface if and only if $\sum _ { j = 1 } ^ { 3 } | \omega _ { j } | ^ { 2 } \neq 0$ on $M$.

The second fundamental form of $X ( M )$ can be expressed as

\begin{equation*} \operatorname{II}( W , V ) = - \operatorname { Re } ( \eta ( W ) d g ( V ) ). \end{equation*}

Moreover, $W$ is an asymptotic direction if and only if $\eta ( W ) d g ( W ) \in i \mathbf{R}$, and $W$ is a principal curvature direction if and only if $\eta ( W ) d g ( W ) \in {\bf{R}}$.

The local Weierstrass representation was discovered in the 1860{}s by K. Enneper and K. Weierstrass. R. Osserman gave the general form on a Riemann surface in the 1960{}s, see [a1] for more details.

References

[a1]	R. Osserman., "A survey of minimal surfaces" , Dover (1986)