Suppose that , is the surface with minimal area among those whose boundary coincides with that of . Let be any smooth function on such that it vanishes on the boundary of
Consider the following family of regular parametrized surfaces
for some small . These surfaces can be thought of as being obtained by varying along its normals, while fixing its boundary. Note that
If achieves minimal area, then
If we take
then we have
(as , if is regular). So
which is precisely the necessary condition we want.
Definition. A smooth surface with vanishing mean curvature is called a minimal surface.
Example 1. Let be the graph of , a smooth function on . Then is a minimal surface if
The above equation is called the minimal surface equation.
Example 2. Let us consider the so-called Enneper surface given by
Then it is known that
It is easy to check that
Hence , and Enneper surface is a minimal surface.
Example 3. The catenoid surface given by
is also a minimal surface.