The following problem is cast in general terms:
Suppose I have a 3 dimensional closed surface that encloses a region with a finite volume. Then, a hole in the surface is cut out. The surface is deformable without allowing intersections. Now I want to deform the surface, and even allow cuts of the surface, to close the edges of the hole. The goal is to minimize a functional of the surface akin to the energy of deformation that is based on the amount of stretching and cutting required to close the hole.
What branch of mathematics or computational geometry should I research to learn about (simple!) techniques to tackle these problems?