We minimize a discrete version of the squared mean curvature integral for polyhedral surfaces in three-space, using Brakke's Surface Evolver [Brakke 1992]. Our experimental results support the conjecture that the smooth minimizers exist for each genus and are stereographic projections of certain minimal surfaces in the three-sphere.
"Minimizing the squared mean curvature integral for surfaces in space forms." Experiment. Math. 1 (3) 191 - 207, 1992.