Minimal Surfaces as Entropy-Density Minimizers: A Classical Investigation

This study proposes that minimal surfaces not only minimize area but also minimize a natural entropy-density functional based on the mean curvature squared. By analyzing surfaces such as the catenoid, helicoid, and Enneper surface, we confirm that minimal surfaces achieve zero entropy-density while non-minimal surfaces (such as spheres, ellipsoids, and paraboloids) exhibit significant positive entropy-density. This offers a new structural variational principle for understanding minimal surfaces in differential geometry.