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



minimal surfaces
In $\mathbb{R}^3$ there is always some minimal surface spanning a given connected simple curve (assuming the curve is not too horrible, say Lipschitz). Here surface needs to be broadly understood as being possibly immersed and of arbitrary topology. This can be seen using the machinery of geometric measure theory and the direct method in the calculus of variations (i.e. by minimizing area). Alternatively, since there is a close connection to harmonic mappings in this context, one can always find a minimal surface spanning the curve that is topologically a disk (this is sometimes referred to as the Douglas-Rado solution). This is also constructed by direct methods but here one minimizes Dirichlet energy of the map. 


If you know more about the bounding curve one can get more information about possible minimal surfaces spanning the curve. For instance if the curve lies on the boundary of a convex set then it spans at least one embedded minimal surface.


In general there is no uniqueness (there are in fact an example due to F. Morgan where a given curve bounds a continuum of minimal surfaces). Though if you know that the bounding curve is analytic and of total geodesic curvature less than $4\pi$ then Nitsche shows that the curve bounds a unique minimal disk (I don't remember if it can also bound surfaces of other topological types).


In general I doubt you could get an explicit parameterization for the surface even if the curve is simplicial. Though you might be able to get something if you look at the Enneper-Weierstrass representation.


