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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

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A check if a surface is minimal is to evaluate the mean curvature and check that it vanishes. That does not seem to be the case for your $f(s,t)$. –  J. M. Nov 19 '10 at 16:31
    
Concerning the question about linear maps: there is no reason an arbitrary linear map should preserve any metric properties. Hence the answer is most certainly not. If you restrict to euclidean transformations (i.e., affine transformations of $\mathbb{R}^3$ preserving the distance $|\mathbf{x} - \mathbf{y}|$), then the answer is yes. –  José Figueroa-O'Farrill Nov 19 '10 at 17:01
    
J.M.: Your computation was much faster than mine! I'm getting negative mean curvature on the interior of the 1st quardant for z=xy, so my suspicion was wrong. Jose: Yes, and for dilations and contractions as well, and I see now how in general not: a linear map should be able to contract one principle curvature while leaving the other invarient, hence changing mean curvature. –  AndrewLMarshall Nov 19 '10 at 22:44

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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.

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fyi: curves of curvature $<4\pi$ can bound other surfaces, imagine a curve which effectively goes around the circle twice without intersecting. this curve has small curvature and it bounds a mobius strip –  Otis Chodosh Oct 25 '13 at 22:10

There is a simple answer to your first question. Imagine two circles of the same radius lying in parallel planes so that the segment joining their centers is perpendicular to the planes. There are two minimal surfaces that bound this configuration. One consists of two disks, one in each plane, and the other is a section of a catenoid. By playing around with the radii and distance you can even have two least area surfaces with the same boundary.

Clearly there is not a unique minimal surface spanning the same piecewise linear curve. You could build a curve that bounded both a minimal disc and minimal Moebius band with little difficulty. I read this in a paper in German that was written in the 50's. I can't remember the author. I think you can find the example in Blaine Lawson's book on minimal surfaces.

Also, Peter Hall gave some examples of nonuniqueness in the late 1980's.

There is a Memoir of the AMS written by Dave Hoffman that constructs lots of families of minimal surfaces, using the Weierstrass representation, and counts of dimensions of holomorphic sections of line bundles to build lots of families.

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Thanks for input. The first isn't an answer to an injection of $S^1$ into $\mathbb R^3$, but the second works well. –  AndrewLMarshall Dec 1 '10 at 6:17

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