TITLE: Variation of curvature with respect to immersion?
QUESTION [7 upvotes]: Let $M$ be a smooth surface and let $f: M \to \mathbb{R}^3$ be a family of immersions given by
$$ f(t) = f_0 + tuN_0, $$
where $f_0$ is some initial immersion, $N_0$ is the associated Gauss map, and $u: M \to \mathbb{R}$ is an arbitrary smooth function.  One can easily work out, e.g., the first variation in the induced area form, but I am having a surprising amount of difficulty deriving (or tracking down known expressions for) variations of higher-order quantities.
Question: What are the first variations of the mean curvature $H$ and the Gauss curvature $K$ with respect to the immersion $f$?
Bonus question: What are the first variations of the principal curvatures?
If these results are not immediately available, I am also appreciative of strategies and techniques for obtaining them.  Finally, my apologies if these questions have well-established answers - I am having no luck finding them in the literature!

REPLY [3 votes]: Have a look at this paper. Chapter 4 contains all that you are asking for, in a more general setting.