For a fixed closed smooth manifolds $M^n$ and $N^{n+1}$, two $C^{k,alpha}$ embeddings $f, f’ : M to N$ are said to be equivalent if there exists $varphi in operatorname{Diff}(M)$ such that $f’ = f circ varphi$. Let $mathcal{M}$ be the set of pairs $((f), g)$, where $(f)$ is an equivalence class of CMC embeddings into the Riemannian manifold $(N,g)$ and $g$ belongs to a fixed open set $Gamma$ of $C^q$ metrics. Brian White showed that $mathcal{M}$ is a $C^{q-j}$ Banach manifold modelled on $Gamma$.

Now let $mathcal{A} : mathcal{M} to mathbb{R}$ be the area functional. I am trying to find its critical points. To this end, let $((f),g)$ be given and consider a curve $((f_t), g_t)$ in $mathcal{M}$ such that $((f_0),g_0) = ((f),g)$ and $d/dt ((f_t), g_t) = (X, h)$ at $t = 0$, where $X$ is a vector field along $f$ and $h$ is a symmetric 2-tensor. Then, by the usual variation formulae for the area and volume elements, we arrive at

$$left. frac{mathrm{d}}{mathrm{d}t} rightvert_{t=0} mathcal{A}((f_t), g_t) = – int_M H_{f} g(X, nu) , mathrm{d}A_{f^ast g} + int_M operatorname{tr}_g(h) , mathrm{d}A_{f^ast g},$$

where $H_f$ is the mean curvature of $f : M to N$, $nu$ is a unit normal for $M$ along $f$ and $operatorname{tr}_g(h)$ is just the trace of $h$ with respect to the metric $g$. How do I conclude? What are the critical points of $mathcal{A}$?