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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w1300401.png" /> be a [[Riemann surface|Riemann surface]]. A harmonic conformal mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w1300402.png" /> then defines a [[Minimal surface|minimal surface]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w1300403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w1300404.png" /> (cf. also [[Harmonic function|Harmonic function]]; [[Conformal mapping|Conformal mapping]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w1300405.png" /> be local [[Isothermal coordinates|isothermal coordinates]]; then
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w1300406.png" /></td> </tr></table>
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Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w1300407.png" /> is harmonic,
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Let $M$ be a [[Riemann surface|Riemann surface]]. A harmonic conformal mapping $X : M \rightarrow {\bf R} ^ { n }$ then defines a [[Minimal surface|minimal surface]] in ${\bf R} ^ { n }$, $n \geq 3$ (cf. also [[Harmonic function|Harmonic function]]; [[Conformal mapping|Conformal mapping]]). Let $z = u + i v$ be local [[Isothermal coordinates|isothermal coordinates]]; then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w1300408.png" /></td> </tr></table>
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\begin{equation*} \sum _ { j = 1 } ^ { n } \Bigl( \frac { \partial X _ { j } } { \partial z } \Bigr) ^ { 2 } = 0. \end{equation*}
  
is a holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w1300409.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004010.png" />. Hence any (branched) minimal surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004011.png" /> can be given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004012.png" /> meromorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004013.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004014.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004016.png" /> can be expressed as
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Since $X$ is harmonic,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation*} \omega _ { j } = 2 \frac { \partial X _ { j } } { \partial z } d z \end{equation*}
  
Such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004018.png" /> is well defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004019.png" /> if and only if for any [[Loop|loop]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004021.png" />,
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a1} X ( p ) = \operatorname { Re } \int _ { p _ { 0 } } ^ { p } ( \omega _ { 1 } , \ldots , \omega _ { n } ). \end{equation}
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004023.png" />, one gets a [[Meromorphic function|meromorphic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004024.png" /> and a meromorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004025.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004026.png" />,
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Such an $X$ is well defined on $M$ if and only if for any [[Loop|loop]] $C$ in $M$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004027.png" /></td> </tr></table>
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\begin{equation} \tag{a2} \operatorname { Re } \int _ { C } ( \omega _ { 1 } , \dots , \omega _ { n } ) = ( 0 , \dots , 0 ). \end{equation}
  
On the other hand, given a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004028.png" /> and a meromorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004029.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004030.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004031.png" />, define
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For $n = 3$, one gets a [[Meromorphic function|meromorphic function]] $g$ and a meromorphic $1$-form $ \eta $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\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*}
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004033.png" />. Thus, (a3) together with (a1) defines a minimal surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004034.png" /> and is called the Weierstrass representation of the minimal surface via the Weierstrass data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004035.png" />.
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On the other hand, given a meromorphic function $g$ and a meromorphic $1$-form $ \eta $ on $M$, define
  
The meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004036.png" /> has the geometric meaning that it is the composite of the spherical mapping (or unit normal vector) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004037.png" /> and the [[Stereographic projection|stereographic projection]] from the north pole, where
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\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}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004038.png" /></td> </tr></table>
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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 $( g , \eta )$.
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004039.png" /> is also called the Gauss map of the minimal surface.
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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|stereographic projection]] from the north pole, where
  
The [[First fundamental form|first fundamental form]] and the [[Gaussian curvature|Gaussian curvature]] of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004040.png" /> can be expressed via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004041.png" />,
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\begin{equation*} N = \frac { 1 } { | g | ^ { 2 } + 1 } ( 2 \operatorname { Re } g , 2 \operatorname { Im } g , | g | ^ { 2 } - 1 ) \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004042.png" /></td> </tr></table>
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and $g$ is also called the Gauss map of the minimal surface.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004043.png" /></td> </tr></table>
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The [[First fundamental form|first fundamental form]] and the [[Gaussian curvature|Gaussian curvature]] of the surface $X ( M )$ can be expressed via $( g , \eta )$,
  
Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004044.png" /> is a regular surface if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004046.png" />.
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\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*}
  
The [[Second fundamental form|second fundamental form]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004047.png" /> can be expressed as
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\begin{equation*} K = - \left( \frac { 4 | d g | } { ( 1 + | g | ^ { 2 } ) ^ { 2 } | \eta | } \right) ^ { 2 }. \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004048.png" /></td> </tr></table>
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Hence $X ( M )$ is a regular surface if and only if $\sum _ { j = 1 } ^ { 3 } | \omega _ { j } | ^ { 2 } \neq 0$ on $M$.
  
Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004049.png" /> is an asymptotic direction if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004050.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004051.png" /> is a principal curvature direction if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w130/w130040/w13004052.png" />.
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The [[Second fundamental form|second fundamental form]] of $X ( M )$ can be expressed as
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\begin{equation*} \operatorname{II}( W , V ) = - \operatorname { Re } ( \eta ( W ) d g ( V ) ). \end{equation*}
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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 [[#References|[a1]]] for more details.
 
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 [[#References|[a1]]] for more details.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Osserman.,  "A survey of minimal surfaces" , Dover  (1986)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  R. Osserman.,  "A survey of minimal surfaces" , Dover  (1986)</td></tr></table>

Latest revision as of 16:57, 1 July 2020

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)
How to Cite This Entry:
Weierstrass representation of a minimal surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_representation_of_a_minimal_surface&oldid=13727
This article was adapted from an original article by Yi Fang (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article