*minimal submanifold* if and only if $H=0$.

The equation $H=0$ is a *second order nonlinear PDE*. We can see this explicitly in the following simple case. For a function $f:U\subseteq\R^{n-1}\rightarrow\R$ where $\overline{U}$ is compact, we see that if $N=\text{Graph}(f)\subseteq\R^n$ then the volume of $N$ is given by $$\operatorname{Vol}(N)=\int_U\sqrt{1+|\nabla f|^2}\operatorname{vol}_U.$$ Any sufficiently small variation can be written $F(N,t)=\text{Graph}(f+th)$ for some $h:U\rightarrow\R$, so we can compute $$\begin{aligned}
\frac{\d}{\d t}\operatorname{Vol}(F(N,t))|_{t=0}&=\frac{\d}{\d t}|_{t=0}\int_U\sqrt{1+|\nabla f+t\nabla h|^2}\operatorname{vol}_U\\
&=\int_U\frac{\d}{\d t}|_{t=0}\sqrt{1+|\nabla f|^2+2t\langle\nabla f,\nabla h\rangle+t^2|\nabla h|^2}\operatorname{vol}_U\\
&=\int_U\frac{\langle \nabla f,\nabla h\rangle}{\sqrt{1+|\nabla f|^2}}\operatorname{vol}_U\\
&=-\int_U h\operatorname{div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}}\right)\operatorname{vol}_U.\end{aligned}$$ We therefore see that $N$ is minimal if and only if this vanishes for all $h$. Hence, $\text{Graph}(f)$ is minimal in $\R^n$ if and only if $$\operatorname{div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^2}}\right)=0.$$ We see that we can write this equation as $\Delta f+Q(\nabla f,\nabla^2 f)=0$ where $Q$ consists of nonlinear terms (but linear in $\nabla^2f$). Hence, if we linearise this equation we just get $\Delta f=0$, so $f$ is harmonic. In other words, the minimal submanifold equation is a nonlinear equation whose linearisation is just Laplace’s equation: this is an example of a nonlinear *elliptic* PDE, which we shall discuss further later.

A plane in $\R^n$ is trivially minimal because if $X,Y$ are any vector fields on the 