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minimal surface is a kind of surface with vanishing mean curvature @xcite . as the mean
curvature is the variation of area functional , minimal surfaces include the surfaces minimizing the area with a fixed boundary @xcite .
there have been many literatures on minimal surface in classical differential geometry @xcite .
because of their attractive properties , minimal surfaces have been extensively employed in many areas such as architecture , material science , aviation , ship manufacture , biology and so on . for instance , the shape of the membrane structure , which has appeared frequently in modern architecture , is mainly based on minimal surfaces @xcite .
furthermore , triply periodic minimal surfaces naturally arise in a variety of systems , including nano - composites , lipid - water systems and certain cell membranes @xcite .    in cad systems ,
parametric polynomial representation is the standard form . for parametric polynomial minimal surface
, plane is the unique quadratic parametric polynomial minimal surface , enneper surface is the unique cubic parametric polynomial minimal surface .
there are few research work on the parametric form of polynomial minimal surface with higher degree .
weierstrass representation is a classical parameterization of minimal surfaces .
however , two functions should be specified to construct the parametric form in weierestrass representation . in this paper
, we discuss the answer to the following questions : what are the possible explicit parametric form of polynomial minimal surface of arbitrary degree and how about their properties ?
the proposed minimal surfaces include the classical enneper surface for cubic case , and also have some interesting properties such as symmetry , containing straight lines and self - intersections . according to the shape properties ,
the proposed minimal surface can be classified into four categories with respect to @xmath0 @xmath2 , @xmath1 and @xmath3 .
the explicit parametric form of corresponding conjugate minimal surfaces is given and the isometric deformation is also implemented .
the paper includes five sections .
_ preliminary _ introduces some notations and lemmas . _ main results _ presents the explicit parametric formula of parametric polynomial minimal surface of arbitrary degree .
the next section , _ properties and classification _ presents the corresponding properties and classification of the proposed minimal surfaces .
the following section focuses on corresponding conjugate counterpart of proposed minimal surface . finally , in _ conclusions _
, we summarize the main results .
in this section , we introduce the following two notations .
@xmath4    @xmath5 and @xmath6 have the following properties :    * lemma 1 * [ lemma : deriv ] @xmath7    * lemma 2 * [ lemma : iteration ] @xmath8 lemma [ lemma : iteration ] can be proved by using the following equation : @xmath9
* theorem 1 * [ ref : form ] if the parametric representation of polynomial surface @xmath10 with arbitrary degree @xmath11 , is given by @xmath12 , where @xmath13 then @xmath10 is a minimal surface .
* proof of theorem 1 . * from lemma [ lemma : deriv ]
, we have @xmath14 hence , @xmath15 is harmonic surface .    by using lemma [ lemma : deriv ]
, we have @xmath16 from lemma [ lemma : iteration ] , we have @xmath17 substituting ( 5)(6)(7)(8 ) into ( 4 ) , we can obtain @xmath18 .
similarly , we have @xmath19 hence , @xmath20 is a parametric surface with isothermal parameterization . from @xcite , if a parametric surface with isothermal parameterization is harmonic , then it is a minimal surface . the proof is completed
from _ theorem 1 _ , if @xmath21 , we can get the enneper surface , which is the unique cubic parametric polynomial minimal surface .
it has the following parametric form @xmath22 enneper surface has several interesting properties , such as symmetry , self - intersection , and containing orthogonal straight lines on it . for the new proposed minimal surface
, we can also prove that it has also has these properties .
if @xmath23 , a kind of quintic polynomial minimal surface proposed in @xcite can be obtained as follows @xmath24 where @xmath25 according to the shape properties , the proposed minimal surface in _ theorem 1 _ can be classified into four classes with @xmath0 @xmath2 , @xmath1 , @xmath3 .
* proposition 1 . * in case of @xmath0 , the corresponding proposed minimal surface @xmath15 has the following properties :    * @xmath15 is symmetric about the plane @xmath26 and the plane @xmath27 , * @xmath15 contains two orthogonal straight lines @xmath28 on the plane @xmath29    fig .
1(a ) shows an example of enneper surface , fig .
1 ( b ) shows an example of proposed minimal surface with @xmath30 . the symmetry plane and straight lines of minimal surface in fig . 1 ( b )
are shown in fig . 1 ( c ) and fig . 1 ( d ) .
* proposition 2 . * in case of @xmath2 , the corresponding proposed minimal surface @xmath15 is symmetric about the plane @xmath29 and the plane @xmath27 .
[ fig : cubic ]    \times[-1,1]$].,title="fig:",scaledwidth=110.0%,scaledwidth=100.0% ] + ( a ) ennerper surface    \times[-1,1]$].,title="fig:",scaledwidth=110.0%,scaledwidth=100.0% ] + ( b ) polynomial surface of degree 7     +    \times[-1,1]$].,title="fig:",scaledwidth=110.0%,scaledwidth=100.0% ] + ( a ) symmetry plane    \times[-1,1]$].,title="fig:",scaledwidth=110.0%,scaledwidth=100.0% ] + ( b ) straight lines    fig .
2 ( a ) present an example of proposed quartic minimal surface and the corresponding symmetry planes are shown in fig . 2 ( b ) .
* proposition 3 . * in case of @xmath1 , the corresponding proposed minimal surface @xmath15 has the following properties :    * @xmath15 is symmetric about the plane @xmath26 , the plane @xmath27 , the plane @xmath31 and the plane @xmath32 .
* self - intersection points of @xmath15 are only on the symmetric planes , i.e. , there are no other self - intersection points on @xmath33 , and the self - intersection curve has the same symmetric plane as the minimal surface .
3 ( a ) present an example of proposed quintic minimal surface and the corresponding symmetry planes are shown in fig .
3 ( b ) .
* proposition 4 . * in case of @xmath3 , the corresponding proposed minimal surface @xmath15 is symmetric about the plane @xmath29 and the plane @xmath27 .    for the case of @xmath34 , it has been studied in @xcite .
fig.4 ( a ) present an example of proposed minimal surface with @xmath34 and the corresponding symmetry planes are shown in fig .
4 ( b ) .
[ fig : cubic ]    \times[-1,1]$].,title="fig:",scaledwidth=110.0%,scaledwidth=100.0% ] + ( a ) quartic minimal surface    \times[-1,1]$].,title="fig:",scaledwidth=110.0%,scaledwidth=100.0% ] + ( b ) symmetry plane    [ fig : cubic ]    \times[-1,1]$].,title="fig:",scaledwidth=110.0%,scaledwidth=100.0% ] + ( a ) quintic minimal surface    \times[-1,1]$].,title="fig:",scaledwidth=110.0%,scaledwidth=100.0% ] + ( b ) symmetry plane    [ fig : cubic ]    \times[-1,1]$].,title="fig:",scaledwidth=110.0%,scaledwidth=100.0% ] + ( a ) minimal surface of degree six    \times[-1,1]$].,title="fig:",scaledwidth=110.0%,scaledwidth=100.0% ] + ( b ) symmetry plane     + a ) @xmath35     + b ) @xmath36     + c ) @xmath37     +     + d ) @xmath38     + e ) @xmath39     + f ) @xmath40     +
* definition 1 . *
if two differentiable functions @xmath41 satisfy the cauchy - riemann equations @xmath42 and both are harmonic , then the functions are said to be _
harmonic conjugate_.    * definition 2 . *
if @xmath43 and @xmath44 are with isothermal parameterizations such that @xmath45 and @xmath46 are harmonic conjugate for @xmath47 , then @xmath48 and @xmath49 are said to be _ parametric conjugate minimal surfaces_.    helicoid and catenoid are a pair of conjugate minimal surfaces . for @xmath50 , we can find out a new pair of conjugate minimal surfaces as follows .    * theorem 2 * [ ref : form ] the conjugate minimal surface of @xmath10 has the following parametric form @xmath51 where @xmath52    it can be proved directly by lemma [ lemma : deriv ] . from [ 2 ] ,
the surfaces of one - parametric family @xmath53 are minimal surfaces with the same first fundamental form .
these minimal surfaces are isometric and have the same gaussian curvature at corresponding points .
5 illustrates the isometric deformation between @xmath15 and @xmath54 .
it is similar with the isometric deformation between helicoid and catenoid .
the explicit parametric formula of polynomial minimal surface is presented .
it can be considered as the generalization of enneper surface in cubic case . the corresponding properties and classification of the proposed minimal surface
are investigated .
the corresponding conjugate minimal surface are constructed and the dynamic isometric deformation between them are also implemented .
this work was partially supported by the national nature science foundation of china ( no.60970079 , 60933008 ) , foundation of state key basic research 973 development programming item of china ( no.2004cb318000 ) , and the nature science foundation of zhejiang province , china(no .
y1090718 ) .                  bletzinger k.w .. 1997 .
form finding of membrane structures and minimal surfaces by numerical continuation .
proceeding of the iass congress on structural morphology : towards the new millennium , nottingham .
isbn 0 - 85358 - 064 - 2 , 68 - 75
weierstrass representation is a classical parameterization of minimal surfaces . 
 however , two functions should be specified to construct the parametric form in weierestrass representation . in this paper 
 , we propose an explicit parametric form for a class of parametric polynomial minimal surfaces of arbitrary degree . 
 it includes the classical enneper surface for cubic case . 
 the proposed minimal surfaces also have some interesting properties such as symmetry , containing straight lines and self - intersections . according to the shape properties , 
 the proposed minimal surface can be classified into four categories with respect to @xmath0 @xmath1 , @xmath2 and @xmath3 . 
 the explicit parametric form of corresponding conjugate minimal surfaces is given and the isometric deformation is also implemented .    minimal surface ; parametric polynomial minimal surface ; enneper surface ; conjugate minimal surface