I’ve spent a few posts talking about the theory behind minimal surfaces. So what? Lets actually look at some.

Prior to the 18th century the only known minimal surface was the plane. This changed when Jean Baptiste Meusnier discovered the first non-planar minimal surfaces, the catenoid and the helicoid.

The catenoid may be parametrized as $\textbf{x}(u,v)=(a\cosh(v)\cos(u),a\cosh(v)\sin(u),av)$. This is a surface of revolution generated by rotating the catenary $y=a\cosh(\frac{z}{a})$ about the $z$-axis. It is easily checked that the mean curvature of $\textbf{x}(u,v)$ is zero. Thus, the catenoid is a minimal surface. It can be characterized as the only surface of revolution which is minimal. That is, if a surface of revolution $M$ is a minimal surface then $M$ is contained in either a plane or a catenoid.

the catenoid

The helicoid may be parametrized as $\textbf{x}(u,v)=(a\sinh(v)\cos(u),a\sinh(v)\sin(u),au)$. It is easily checked that the mean curvature of $\textbf{x}(u,v)$ is zero. Thus, the helicoid is a minimal surface. It can be characterized as the only minimal surface, other than the plane, which is also a ruled surface. That is, any ruled minimal surface in $\mathbb{R}^3$ is part of a plane or a helicoid.

the helicoid

Placing geometric restrictions on surfaces is a common theme in classification in the study of minimal surfaces. For example, assuming a surface has a parametrization of the form $\textbf{x}(u,v)=g(u)+h(v)$, one can explicitly solve the resulting differential equation to find the minimal surface solution $f(x,y)=\frac{1}{a}\ln (\frac{\cos ax}{\cos ay})$, which locally parametrizes Scherk’s minimal surface (discovered by Heinrich Ferdinand Scherk in 1835). Note that although this surface can be realized locally as a graph, its domain of definition is not the entire plane as it must be represented on patches of the form $-\frac{\pi}{2}<\frac{\pi}{2}$ and –$\frac{\pi}{2}<\frac{\pi}{2}$.

Scherk's surface