|The study of area-minimizing surfaces in 3-space goes back to Euler. Interest in higher dimensions is, of course, more recent. After reviewing some of the history of area-minimizing surfaces in 3-space and some of the resulting examples of minimal surfaces, we will describe some of the difficulties involved in finding area-minimizing hypersurfaces computationally.
The least gradient method is a computational scheme for finding an approximation to a globally area-minimizing oriented hypersurface having a given boundary. The least gradient method avoids the difficulties alluded to above. The trade-off is that to obtain the best results and the best understanding the given boundary curve must lie on the surface of a convex body. Examples of applying the least gradient method will be shown.
Time permitting, we will discuss more recent work applying and extending the least gradient method to boundary curves that do not necessarily lie on the surface of a convex body.