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    $\begingroup$ Very nice question. Thanks for taking the time ask it and write this up so carefully. +1 $\endgroup$ Commented Jan 21, 2015 at 17:39
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    $\begingroup$ "we can slide around the parametrization" It is diffeomorphism :-) $\endgroup$ Commented Jan 22, 2015 at 2:18
  • $\begingroup$ My two cents.The catenoid/helicoid isometric morphing preserves Gauss curvature K and zero mean curvature H as a special minimal surface case. As one workaround instead of starting with given closed boundary of disc and attempting to find the minimal surface spanned in it in direct computation, it would be perhaps insightful to take arbitrary closed loops written on the catenoid, find minimal surface using a Mathematica FEM algorithm and to directly verify with known solution. contd $\endgroup$ Commented Jan 22, 2015 at 19:31
  • $\begingroup$ This way it allows one to discover or formulate a certain relationship among differentials with functional relationships that allow generalization to advantage into other cases also. Using physical soap films spanning inside loops and using holographic optical methods is another easy experimental verification method. $\endgroup$ Commented Jan 22, 2015 at 19:38
  • $\begingroup$ @Narasimham: Sounds like a good idea. And with ybeltukov's code, you can now try it! Let me know if you find anything interesting :) $\endgroup$ Commented Jan 22, 2015 at 19:56