Let $M^3$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the normal variation $\phi: \Sigma \times \Bbb{R}\to M$ given by
$$\phi(p,t)=\exp_p(tN(p)),$$
when $N$ is a normal vector field along to $\Sigma.$
I want to show that:
Lemma: There is $\delta>0$ such that $\phi:\Sigma\times [0,\delta) \to M$ is a embedded.
I tried a sketch with tubular neighborhood but I don't conclude anything in this direction. In fact, I know that there is the tubular neighborhood of $\Sigma$, but I don't how immerse this in $M$.
Anyone has a little help?
Thanks so much.
Let $v$ be any tangent vector of $\Sigma$, then $d\phi_{(p, 0)}(v, 0)=v$ since when $t=0$, $\phi(p, 0)=p$ is the projection to $\Sigma$. And $d\phi_{(p, 0)}(0, dt)=N_p$ based on the definition of the $\exp$ map.
So if we choose a basis on $T_p\Sigma$ together with $d/dt$ for $T_p\Sigma\times T\mathbb{R}$ and the same basis together with $N$ in $T_p M$, the differential is of the form of an identity matrix. Thus it is an embedding when $t=0$.
The conclusion follows immediately since the map is smooth.