plane then $\nabla_X^{\perp}Y=0$ as the second fundamental form of a plane is zero.

For curves $\gamma$, $H=0$ is equivalent to the geodesic equation $\nabla_{\dot{\gamma}}\dot{\gamma}=0$.

The most studied minimal submanifolds (other than geodesics) are minimal surfaces in $\R^3$, since here the equation $H=0$ becomes a scalar equation on a surface, which is the simplest to analyse. In general we would have a system of equations, which is more difficult to study.

The helicoid $M=\{(t\cos s,t\sin s,s)\in\R^3\,:\,s,t\in\R\}$ is a complete embedded minimal surface, discovered by Meusnier in 1776.

The catenoid $M=\{(\cosh t\cos s,\cosh t\sin s ,t)\in\R^3\,:\,s,t,\in\R\}$ is a complete embedded minimal surface, discovered by Euler in 1744 and shown to be minimal by Meusnier in 1776. The catenoid is another explicit example which is a critical point for volume but not minimizing.

In fact the helicoid and the catenoid are locally isometric, and there is a 1-parameter family of locally isometric minimal surfaces deforming between the catenoid and helicoid: see, for example, [@Gray Theorem 16.5] for details.

It took about 70 years to find the next minimal surface, but now we know many examples of minimal surfaces in $\R^3$, as well as in other spaces by studying the nonlinear elliptic PDE given 