Is there a unique minimal surface with boundary $S_2$?

Question

Take a loop of wire ($S_2$), twisted into a complicated shape. One can dip the wire in soap to form a minimal surface soap film.

Is there a unique minimal surface or can there be more than one inquivalent surface with minimal area?

Answer 1

There can be more than one with minimal area. Here's an example:

Wrap a piece of wire around two posts stuck in a board, so there's a half-circle, a long straight section, another half-circle, and another long straight piece.

Remove the posts from the board, and lay one of them down across the middle of your loop, so that from overhead you see this:

           ****
/----------****---------\
|          ****         |
\----------****---------/   
           ****

where the stars are the post, and the stuff made of up slashes are my best attempt at making semicircles.

Holding down the post, lift up each semicircle until the four "straight" sections are vertical.

The highly-symmetric shape you get has three "extremal" surfaces. Two look like tongue depressors with a 180-degree arcing bend in the middle; the third looks like the "average" of these two. The first two are equal area minimal surfaces (at least if the "straight bits" are long enough; the third is saddle like and is "extremal" in some way I can't precisely remember.

If you shorten the straight-line segments, you get to a point where the two "flatter" surfaces both collapse (if they're soap-films) to the central potato-chip-like one, which is then a unique minimal surface.