I have a few really basic questions about minimal surfaces. Does a smooth or piecewise smooth injection $S^1$ into $\mathbb R^3$ always give a unique minimal surface or are there instances with discrete distinct solutions? Can it not be the case that a 1-parameter family of minimal surfaces exists for a given "frame"? Do linear maps preserve minimal surfaces? I'm guessing no, but I don't have a good example in mind.

Also, if I have a simplicial decomposition of $S^1$ and map it into $\mathbb R^3$ with a simplicial map, is it known that there is a unique minimal surface spanning this piecewise linear frame? Is the formula readily given?

These questions came up defining a surface $f(s,t)=(1-s)(1-t)v_0+(1-s)tv_1+s(1-t)v_3+stv_4$ to interpolate the simplicial map defined by $f(0,0)=v_0, f(0,1)=v_1, f(1,1)=v_2, f(1,0)=v_3$ which maps $\partial I^2$ into $\mathbb R^3$. I believe this surface is minimal, and wonder how this works for polygons with more vertices than the square.

Thanks