restart: with(plots): with(linalg):

with(plots):with(liesymm): setup(u,v):

Phi:=z^(3);

F(z):=subs(z=u+I*v,Phi);

phi(u,v):=evalc(Re(F(z)));

psi(u,v):=evalc(Im(F(z)));

X:=[u,v,phi(u,v),psi(u,v)];

Xu:=diff(X,u);

Xv:=diff(X,v);

E:=simplify(dotprod(Xu,Xu));

F:=simplify(dotprod(Xu,Xv));

G:=simplify(dotprod(Xv,Xv));

Xuu:=diff(Xu,u);

Xuv:=diff(Xu,v); Xvv:=diff(Xv,v);

H:=simplify((1+G)*(Xuu[3]+Xuu[4])+(1+E)*(Xvv[3]+Xvv[4])-2*F*(Xuv[3]+Xuv[4]));

restart: with(plots): with(linalg):

F(tau):=1/(2*tau^2);

X:=((int((1-tau^2)*F(tau),tau)));

Y:=((int(I*(1+tau^2)*F(tau),tau)));

Z:=((int(2*tau*F(tau),tau)));

X:=subs(tau=exp(z),X); Y:=subs(tau=exp(z),Y); Z:=subs(tau=exp(z),Z);

X:=subs(z=u+I*v,X); Y:=subs(z=u+I*v,Y); Z:=subs(z=u+I*v,Z);

X1Re:=simplify(factor(evalc(Re(X))),trig);

X1Im:=simplify(factor(evalc(Im(X))),trig);

X2Re:=simplify(factor(evalc(Re(Y))),trig);

X2Im:=simplify(factor(evalc(Im(Y))),trig);

X3Re:=simplify(factor(evalc(Re(Z))),trig);

X3Im:=simplify(factor(evalc(Im(Z))),trig);

RRR:=evalm([X1Re,X2Re,X3Re]); RRI:=[X1Im,X2Im,X3Im];

EI:=factor(diff(RRI,u));BI:=factor(diff(RRI,v));

E:=diff(RRR,u);B:=diff(RRR,v);

EE:=innerprod(E,E):BB:=innerprod(B,B):Poincare1:=EE-BB;Poincare2:=innerprod(E,B);

a:=1;b:=1;

plot3d(RRR,u=-1.3..1.3,v=-1.3..1.3,orientation=[134,86],numpoints=5000,style=PATCH,title=`Real part - Mandelbrot Minimal Surface`);

Next, use the Frame method to compute the Mean curvature in terms of the position vector.

The position vector in R3

RR:=[xx,yy,zz];

Yu:=diff(RR,u);

Yv:=diff(RR,v);

NNU:=crossprod(Yu,Yv);

Scale the adjoint normal field here by rho

rho:=innerprod(NNU,NNU)^(1/2);

The choice of the scaling factor rho will determine the torsion coefficients of the system. As the position vector is paramtrically defined, the 1-form, small omega, vanishes identicall, and therefore there is no affine (translational dislocation shear) torsion for the minimal surfaces. However, the rotational shear disclination torsion coefficients exist, except for the scaling factor rho chosen above. The Mean curvature of the Holomorphic curves vanishes for all scaling factors.

rho:=1;

This vector (surface normal) NNU can be computed from the Adjoint Matrix operation on the two

tangent vectors Yu and Yv. The basis frame utilizes this surface normal with arbitrary scaling

NN:=([factor(NNU[1]),factor(NNU[2]),simplify(factor(NNU[3]))]);

FF:=array([[Yu[1],Yv[1],NN[1]/rho],[Yu[2],Yv[2],NN[2]/rho],[Yu[3],Yv[3],NN[3]/rho] ] );

The Repere Mobile or FRAME MATRIX, FF. note that the frame matrix is not orthonormal.!!

detFF:=simplify((det(FF)));

For the case at hand, the deteminant is non-zero globally, hence an inverse always exists.

FFINVD:=evalm(FF^(-1));

The 1-form components of the differential position vector with respect to the Basis Frame, F.

dR:=innerprod(FFINVD,[d(xx),d(yy),d(zz)]);

sigma1:=(wcollect(dR[1]));

sigma2:=wcollect(dR[2]);

Note that sigma1 is du and sigma2 is dv for a parametric Monge surfaces!!

omega:=(wcollect(dR[3]));

Note that this term vanishes for a parametric Monge surface, hence parametric Monge surfaces exhibit no TORSION!! of the Affine type ( that is there is no translational shear defects!)

Compute the Cartan Matrix of connection forms from C=[F(inverse)] times d[F]

dFF:=array([[d(FF[1,1]),d(FF[1,2]),d(FF[1,3])],[d(FF[2,1]),d(FF[2,2]),d(FF[2,3])],[d(FF[3,1]),d(FF[3,2]),d(FF[3,3])]]);

cartan:=evalm(FFINVD&*dFF):

The interior connection coefficients (can be Christoffel symbols on the parameter space

Gamma11:=(wcollect(cartan[1,1]));

Gamma12:=(wcollect(cartan[1,2]));

Gamma21:=(wcollect(cartan[2,1]));

Gamma22:=(wcollect(cartan[2,2]));

The second fundamental form or shape matrix comes from the third row of the Cartan matrix

h1:=wcollect(cartan[3,1]);

gamma1:=wcollect(cartan[1,3]);

h2:=(wcollect(cartan[3,2]));

gamma2:=(wcollect(cartan[2,3]));

Next compute the expansion_twist:

The abnormality for the parametric surface will show up as a non-zero entry in the [3,3] slot of the Cartan Matrix. The abnormaility is always an exact differental for parametric and Monge surfaces. The expansion_twist (Big Omega) is not zero unless the euclidean scaling is subsumed.

Therefore implicit Monge surfaces will admit disclination defects (Torsion of the second kind due to rotations)

Omega:=(wcollect(factor((cartan[3,3]))));

(Omega vanishes for the euclidean type of normalization.

shape11:=-factor(gamma1&^d(v)/d(u)&^d(v));

shape12:=-factor(gamma1&^d(u)/d(v)&^d(u));

shape21:=-factor(gamma2&^d(v)/d(u)&^d(v));

shape22:=-factor(gamma2&^d(u)/d(v)&^d(u));

SHAPE:=array([[shape11,shape12],[shape21,shape22]]):

HH:=simplify(trace(SHAPE)/2):

print(`Mean Curvature is `,factor(HH));

KK:=simplify(det(SHAPE)):

print(`Gauss Curvature is `,factor(KK));

xxi;yyi;zzi;

plot3d([xxi,yyi,zzi],u=-1.3..1.3,v=-1.3..1.3,orientation=[134,86],numpoints=5000,style=PATCH,title=`Imag part - Mandelbrot Minimal Surface`);

Mandelbrodt minimal surface.