Another Minimal Torus With One End

<<Own`Mesh`

Weierstrass Representation

phi2[z_]:=1/((Sqrt[z])(Sqrt[(z^2-1)]));
phi1[z_]:=z(z^2-1)/((Sqrt[z])(Sqrt[(z^2-1)]));

G[z_]:=1/((Sqrt[z]) (Sqrt[-1+(z^2)]));

dh[z_]:=1;

rho=Sqrt[((8 Gamma[(5/4)] Gamma[(9/4)])/((Gamma[(3/4)])^2))]//N

ω2[z_] :=I(rho phi1[z]+phi2[z]/rho )/2;
ω1[z_] :=( rho phi1[z]-phi2[z]/rho)/2;
ω3[z_]:=dh[z]

cp=1.+I;

δ=Re[{
		NIntegrate[ω1[z],{z,cp,0}],NIntegrate[ω2[z],{z,cp,0}],
	0}]

f[w_]:=Re[{
		NIntegrate[ω1[z],{z,cp,w}],NIntegrate[ω2[z],{z,cp,w}],
	w}]-δ;

n[z_]:=StereographicProjection[1/rho G[z]]

ϵ=10^-8.;

Setting up the mesh

r1=1.2;
nr1=6;
nr2=2;
nt=8;

dm=RectangularDomain[NRange[0,1,nr1]⋃NRange[1,r1,nr2],NRange[0,Pi/2.,nt]];

ϵ=0.00000001;
polar[z_]:=Re[z]E^(I Im[z])+ϵ(1+I)

dm2=MeshApply[polar,dm];

Drawing a surface patch

p1=MeshPlot3D[{f[x+I y],n[x+I y]},{x,y},dm2];
p2=MeshJoin[p1,MeshRotate[p1,StraightLine[{0,0,0},f[I]]]];
p3=MeshJoin[p2,MeshReflect[p2,Plane[{0,1,0},0]]];
p4=MeshJoin[p3,MeshReflect[p3,Plane[{1,0,0},0]]];

gr=Mesh3DToGraphics3D[p4];

Show[gr]