Reference: Here
At page 15, we could see that the first variation of the induced metric on a hypersurface $\Sigma(t)$ computed as
$$\frac{\partial}{\partial t}g_{\Sigma(t)} = v(t) g_{\Sigma(t)}, \forall t \in (-\delta,\delta) \tag{1}$$
since $\Sigma(t)$ is umbilic and $H_t$ is constant, where $v$ is a real function.
And then for all $t \in (-\delta, \delta)$
$$g_{\Sigma(t)} = e^{\int^t_0 v(s) ds} = u(t)^2 g_{\mathbb S^2} \tag{2}$$
where $u(t)=a e^{\int^t_0 v(s) ds}$ with $a^2=|\Sigma|/4\pi \in (0,1)$.
Question:
Where do I start to get the expression (1) and (2)? I don't have any clue at all. Thank you.