1.4. Minimal surfaces 15 x y a b ya yb A B Figure 1.8. Profile curve 1992 Oprea, 2000). (For a closed soap bubble, without fixed bound- aries, excess air pressure within the bubble prevents the surface area of the bubble from shrinking to zero.) Euler (1744) discovered that the catenoid, the surface generated by a catenary or hanging chain (see Figure 1.9), minimizes surface area. As you doubtless know, however, from playing with soap films, if you pull two parallel hoops too far apart, the catenoid breaks, leav- ing soap film on the hoops. This was first shown analytically by Goldschmidt (1831). For two parallel, coaxial hoops of radius r, the area of a catenoid is an absolute minimum if the distance between the hoops is less than 1.056 r. This area is a relative minimum for distances between 1.056 r and 1.325 r. For distances greater than 1.325 r, the catenoid breaks and the solution jumps to the discontin- uous Goldschmidt solution (two disks). Joseph Lagrange (1762) then proposed the general problem of finding a surface, z = f(x, y), with a closed curve C as its boundary, that has the smallest area. That is, we now wish to minimize a double

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