Surfaces with $0$ Mean curvature (definition and proof that minimal surfaces are critical points of the Area!)

Examples

• Catenoid
• Helicoid

We sitll haven't been able to classify minimal surfaces (2017)

Applications in physical systems

Soap membranes (and other bilayer film systems) will minimize their surface in equilibrium to minimize Surface tension energy.

In general given a boundary, given by a closed Curve, is there a minimal surface with that boundary in R^3? In 1930s, it was proven that yes.

See the Parsimonious universe book.