Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$, let $DN_t : H^{1/2}(\partial M) \to H^{-1/2}( \partial M)$ be the Dirichlet-to-Neumann map in the Riemannian manifold $(M, g_t)$, i.e., $DN_t(u) = \frac{\partial \widehat{u}}{\partial \nu_t}$, where $\widehat{u}$ is the harmonic extension of $u$ and $\nu_t$ is the outward unit normal to $\partial M$ with respect to $g_t$.
Here is my question: is it true that, for any $u \in H^{1/2}(\partial M)$, the map $t \mapsto DN_t(u)$ is analytic in $t$?