I need to investigate the properties of open curves which evolve according to the standard curve-shortening flow (Wikipedia link), but with fixed extremes as boundaries (si it should converge to the geodesic between such two points?). I have not found much literature on the subject so I decided to try to solve initially the problem numerically (with Pyhton as I am more familiar with it). Any ideas/suggestions would be massively appreciated. Thank you
Anthony Carapetis has code on his website that runs curve shortening flow. You can find that at http://a.carapetis.com/code and it links to a github account which has the source code. His demonstration is for closed curves though.
The issue with open curves is that there is not going to be a unique solution if you don't impose some sort of boundary conditions. This issue can be avoided if you restrict your attention to infinite curves, but you still need to be a little careful to ensure uniqueness and existence. If you are looking for a reference on infinite curves I would suggest looking st the work of Sigurd Angenent. Also there is a paper Self-similar solutions to the curve shortening flow by Hoeskuldur Halldorsson which classifies all of the self similar planar solutions. This includes quite a few infinite curves.