$X$ is an admissible variation of $\mathbf{x}$ I'm reading this subject as a hobby. Could someone help me solve this problem, since I have been doing some geometry for some time?

Let $\mathbf{e}_{3}$ be  smooth unit normal along the immersion $\mathbf{x} \colon M \to \mathbb{R}^3$ compatible with the orientation of $M$. 
If $g$ is a smooth function with compact support $S\subset M$ then there
  exist $\epsilon >0$ such that
  $$
         X
  \colon M \times (-\epsilon, \epsilon)
  \to    \mathbb{R}^3,
  \quad  X(m,t)
  =      \mathbf{x}(m) + t g(m) \mathbf{e}_3(m)
$$
  is an admissible variation of $\mathbf{x}$.
  If $a$ and $c$ are the principal curvatures of $\mathbf{x}$, then $\epsilon = \min_{overS}\left\{ \frac{1}{|a|},\frac{1}{|c|}\right\} $ works.

The definition of admissible variation is as follows:

Definition 8.1.
  An admissible variation of $\mathbf{x}$ is any smooth map
  $$
         X
  \colon M \times (-\epsilon, \epsilon)
  \to    \mathbb{R}^3,
$$
  with compact support, such that for each $t \in (-\epsilon, \epsilon)$, the map
  $$
         \mathbf{x}_t
  \colon M
  \to    \mathbb{R}^3,
  \quad  \mathbf{x}_t(m)
  =      X(m,t),
$$
  is an immersion.
  The support of $X$ is the closure in $M$ of the set of points of $M$ where $\mathbf{x}_t(m) \neq \mathbf{x}(m)$, for some $t$.
(Original scanned image here.)


This would be correct if you had no function $g$ in the variation, i.e., if $g=1$ everywhere. You'll need to divide your $\epsilon$ by the maximum of $|g|$ on $M$. Let's call that quantity $C$.
Then the result is easy enough to prove. To simplify things, assume there are no umbilic points and let $\mathbf e_1,\mathbf e_2$ be a principal moving frame, with dual coframe $\omega_1,\omega_2$. Then $d\mathbf e_3 = -(k_1\omega_1\mathbf e_1 + k_2\omega_2\mathbf e_2)$, and 
\begin{align*}
d\mathbf x_t &= d\mathbf x + tg d\mathbf e_3 + t\,dg\,\mathbf e_3 \\
&= (1-tgk_1)\omega_1\,\mathbf e_1 + (1-tgk_2)\omega_2\,\mathbf e_2 + t\,dg\,\mathbf e_3.
\end{align*}
This will have rank $2$ (independent of the nature of the function $g$) provided $|t|<\dfrac 1{C\sup|k_i|}$ for $i=1,2$.
