Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $k=0,1,2, \dots$ be the eigenvalues of the Laplacian $\Delta_{g_t}$ acting on $C^2(M)$, listed with multiplicities. For each $k = 0, 1, 2, \dots$, let $E_k(t)$ be the eigenspace associated with $\lambda_k(t)$.
For some fixed $k \geq 1$, let $j \leq k$ be such that $\lambda_{j}(0) = \lambda_k(0)$ and $\lambda_{j-1}(0) < \lambda_j(0)$.
Let $$\mathfrak{F}(t) = \bigcup_{i=0}^{j-1} E_i(t).$$
In a paper by Fraser and Schoen (Minimal surfaces and eigenvalue problems, Lemma 2.2), they claim that $\mathfrak{F}(t)$ has constant dimension for small $t$. Could you help me understand why this is true?