- View Source( Dr. Robert Bryant's work in Finsler space introduced me

to this sort of geometry: work in Calabi-Yau surfaces

and reading a paper of Dr. Barth's on K3 surfaces

inspired this.)

Here is my idea of a definition of the cubic analog to the Weierstrass

minimal surface

in a Finsler cubic "field':

Let M be a Finsler manifold associated with a cubic

(M,ds^3) such that they are an orientated Finsler surface.

Then any point P an element of M is a topological neighborhood

which can be parametrized in terms of isothermal parameters.

Call N: M->S^2 it's Descartes map.

Denote H: M->Real as the mean curvature associated with map N

such that the analog:

deltaX=3*H*N

Then the (M,ds^3) produces Harmonic 1-forms on M of

Sum[dX(i)^3,{i,1,3}]=0

In terms like those traditional to Weierstrass minimal surfaces

in this cubic analog:

phi1(f,g)=3*f*g/(1+g^3)

phi2(f,g)=3*f*g^/(1+g^3)

phi3(f,g)=-3*f*g/(1+g^3)^(2/3)

Mathematica:

phi1[f_, g_] = 3*f*g/(1 + g^3)

phi2[f_, g_] = 3*f*g^2/(1 + g^3)

phi3[f_, g_] = -3*f*g/(1 + g^3)^(2/3)

FullSimplify[phi1[f, g]^3 + phi2[f, g]^3 + phi3[f, g]^3]

In the simple f=1; g=t folium case

I have integrated a path on the Manifold M in Mathematica.

In three parts the note book is:

Part one folium 3d projected on triazial "tube" cyclinder:

Clear[x, y, x, t, f, g, h, s, g0]

x = 3*t/(1 + t^3)

y = 3*t^2/(1 + t^3)

z = -3*t/(1 + t^3)^(2/3)

FullSimplify[x^3 + y^3 + z^3]

ParametricPlot3D[{x, y,

Re[z]}, {t, -3, 3}, Axes -> False, Boxed -> False, PlotPoints -> 1000]

g0 = ParametricPlot3D[{x*Cos[p], y*Cos[p + 2*Pi/3], Re[z]*Cos[p +

4*Pi/3]}, {p, 0, Pi}, {t, 0.01, 4}, Axes -> False, Boxed -> False,

PlotPoints -> {40, 40}]

ga = ParametricPlot3D[{x*

Cos[p], y*Cos[p + 2*Pi/3],

Re[z]*Cos[p + 4*Pi/3]}, {p, 0, Pi}, {t, -0.99, -0.01}, Axes ->

False, Boxed -> False, PlotPoints -> {40, 40}]

Show[g0, ViewPoint -> {-0.061, 3.381, - 0.127}]

Show[{g0, ga}, ViewPoint -> {-0.061, 3.381, - 0.127}]

Part two (with Mathematica specific characters)

the intgrations:

\!\(f[s_] = Integrate[x, {t, \(-3\), s}]\n

f1[s_] = FullSimplify[3\ \((3 + s)\)*\((\(2\ \[ImaginaryI]\

? + 2\ \@3\ ArcTan[7\/\@3] -

2\ \@3\ ArcTan[\(1 - 2\ s\)\/\@3] - Log[13\/4] - 2\ Log[1 + s] + Log[1 \

- s + s\^2]\)\/\(6\ \((3 + s)\)\))\)]\n

g[s_] = Integrate[y, {t, \(-3\), s}]\n

g1[s_] = FullSimplify[3\ \((3 + s)\)*\((\(\(-\[ImaginaryI]\)\ ? - Log[26] + \

Log[1 + s\^3]\)\/\(3\ \((3 + s)\)\))\)]\n

h[s_] = Integrate[z, {t, \(-3\), s}]\n

h1[s_] = FullSimplify[3\ \((3 +

s)\)*\((\(9\ \((\(-1\))\)\^\(2/3\)\ Hypergeometric2F1[

2\/3, 2\/3, 5\/3, 27] + s\ \((\(-\(s\^3\/\((1 +

s\^3)\)\^2\)\))\)\^\(1/3\)\ \((1 + s\^3)\)\^\(2/3\)\ Hypergeometric2F1[

2\/3, 2\/3, 5\/3, \(-s\^3\)]\)\/\(2\ \((3 + s)\)\))\)]\)

Part three the poltting of the new surface:

ParametricPlot3D[{Re[f1[s]], Re[g1[s]], Re[h1[s]]}, {s, -10, 10}, Axes -> \

False, Boxed -> False, PlotPoints -> 1000]

gb = ParametricPlot3D[{Re[f1[s]]*Cos[p], Re[g1[s]]*Cos[p + 2*

Pi/3], Re[h1[s]]*Cos[p + 4*Pi/3]}, {p, 0,

Pi}, {s, 0.01, 4}, Axes -> False, Boxed -> False, PlotPoints -> {40, 40}]

gc = ParametricPlot3D[{Re[f1[s]]*Cos[p], Re[g1[s]]*Cos[p + 2*Pi/3], Re[h1[

s]]*Cos[p + 4*Pi/3]}, {p, 0,

Pi}, {s, -0.99, -0.01}, Axes -> False, Boxed -> False, PlotPoints \

-> {40, 40}]

gd = ParametricPlot3D[{Re[f1[s]]*Cos[p], Re[g1[s]]*Cos[p + 2*Pi/

3], Re[h1[s]]*Cos[p + 4*Pi/3]}, {p, -Pi, 0}, {s, 0.01, 4},

Axes -> False, Boxed -> False, PlotPoints -> {40, 40}]

ge = ParametricPlot3D[{Re[f1[s]]*Cos[p], Re[g1[s]]*Cos[p + 2*Pi/3],

Re[h1[s]]*Cos[p + 4*Pi/3]}, {

p, -Pi, 0}, {s, -0.99, -0.01}, Axes -> False, Boxed -> False,

PlotPoints -> {40, 40}]

Show[{gb, gc, gd, ge}, ViewPoint -> {-0.061, 3.381, - 0.127}]

Show[{gb, gc, gd, ge}]

Picture of the surface is attached.

This plain simple surface is just an indication of

other possible surfaces by conformal mapping functions of (f,g)

onto the Descartes map and integrating.

That is a Finsler type manifold with a 1/r^3 force.

It gives a strange twisted torus of a negative curvature like

a pseudosphere with a hole in it that has a perpendicular

pseudosphere to it.

This is the K3 like case of a Finsler manifold with the

folium loop in it.

It is probably closer to what a real Calabi-Yau / Null-Ricci

for these non-Riemannian geometries would look like.

Respectfully, Roger L. Bagula

11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :http://www.geocities.com/rlbagulatftn/Index.html

alternative email: rlbagula@...