School “An Invitation to Geometry and Topology via $\operatorname{G}_2$” at Imperial College London in July 2014.

Minimal submanifolds
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We start by analysing the submanifolds which are critical points for the volume functional. Let $N$ be a submanifold (without boundary) of a Riemannian manifold $(M,g)$ and let $F:N\times(-\epsilon,\epsilon)\rightarrow M$ be a variation of $N$ with compact support; i.e. $F=\text{Id}$ outside a compact subset $\overline{S}$ of $N$ with $S$ open and $F(p,0)=p$ for all $p\in N$. The vector field $X=\frac{\partial F}{\partial t}|_N$ is called the variation vector field (which will be zero outside of $\overline{S}$). We then have the following definition.

$N$ is *minimal* if $\frac{\d}{\d t}\operatorname{Vol}(F(S,t))|_{t=0}=0$ for all variations $F$ with compact support $\overline{S}$ (depending on $F$).

Notice that we do not ask for $N$ to minimize volume: it is only stationary for the volume. It could even be a maximum!

A plane in $\R^n$ is minimal since any small variation will have larger volume.

Geodesics are locally length minimizing, so geodesics are minimal. However, as an example, the equator in $\mathcal{S}^2$ is minimal but not length minimizing since we can deform it to a shorter line of latitude.

For simplicity let us suppose that $N$ is compact. We wish 