Let $M^3$ be a closed orientable smooth manifold and let $g_t$ be a (smooth) $1$-parameter family of Riemannian metrics on $M$, $t \in \mathbb{R}$. Let $P \subset M$ a fixed closed orientable embedded surface with the property that $P$ is minimal with respect to the metrics $g_t$. Finally, let $\Sigma_t$, $t \in \mathbb{R}$, be a (smooth) $1$-parameter family of embedded minimal normal graphs over $P$ (with respect to $g_t$); this means that there exists a smooth family of maps $f_t : P \to \mathbb{R}$ such that
$$\Sigma_t = \{ \operatorname{exp}_p(f_t(p)N_t(p)) : p \in P \}$$
where $N_t$ is a unit normal for $P$ with respect to $g_t$ and each $\Sigma_t$ is minimally embedded in $(M, g_t)$.
My question: if $\Sigma_0 = P$ and $\Sigma_t \neq P$ for $t \neq 0$, how does the number of connected components of $Q_t := P \setminus \Sigma_t$ change with $t \neq 0$? Is it constant?
My intuition tells me that if the number of connected components in $Q_t$ changes at $t = t_0$ then $\Sigma_{t_0}$ and $P$ must be tangent, with one surface locally on one side of the other, which is forbidden by the maximum principle. Is this reasoning correct?