Find initial surface to minimize between two close curves

Question

I have two close curves in space defined by $g$ and $h$ with:

Px[t_, A_, p_] := ((\[Pi] - t) t (p \[Pi] (\[Pi] - 2 t)^2 + 16 A (\[Pi] - t) t))/\[Pi]^4
X[t_, a_, px_] := If[t <= \[Pi], Px[t, a, px], -Px[t - \[Pi], a, px]]

Py[t_, A_, \[Epsilon]_] := ((\[Pi] - 2 t) (256 A (\[Pi] - t)^2 t^2 (9 \[Pi]^2 - 16 \[Pi] t + 16 t^2) + (\[Pi] - 4 t)^2 (3 \[Pi] - 4 t)^2 (3 \[Pi]^2 + 38 \[Pi] t - 38 t^2) \[Epsilon]))/(27 \[Pi]^7)
Y[t_, b_, \[Epsilon]_] := If[t <= \[Pi], Py[t, b, \[Epsilon]], Py[2 \[Pi] - t, b, \[Epsilon]]]

Pz[t_, A_, p_] := ((\[Pi] - 2 t) (\[Pi] - t) t (3 p \[Pi] (\[Pi] - 4 t)^2 (3 \[Pi] - 4 t)^2 + 256 A (\[Pi] - t) t (9 \[Pi]^2 - 16 \[Pi] t + 16 t^2)))/(27 \[Pi]^7)
Z[t_, c_, pz_] := If[t <= \[Pi], Pz[t, c, pz], Pz[t - \[Pi], c, pz]]

g[t_] := {X[t, 2, 1], Y[t, 1, .1], Z[t, 1, 2]}
h[t_] := {-Y[t, 1, .1], X[t, 2, 1], Z[t, 1, 2]}

ParametricPlot3D[{g[t], h[t]}, {t, 0, 2 \[Pi]}]

I would now want to build the minimal surface defined by those boundaries. But I have trouble adapting this solution to a situation where the initial surface is not a Disk[] but an Annulus[].

Is it easy to adapt, or do I miss something topologically sneaky?
How do I construct this minimal surface?

Answer 1

You can get an initial surface with an easy interpolation:

Show [ParametricPlot3D[{g[t], h[t]}, {t, 0, 2 \[Pi]}], 
ParametricPlot3D[u g[t] + (1 - u) h[t] , {t, 0, 2 \[Pi]}, {u, 0, 1}],Axes -> False, Boxed -> False]