Suppose our ambient manifold is a path-connected planar region in $\mathbb{R}^2$. Define $g_z = (1 + (z - s))^2dx^2 + dy^2, s\in [a,b]$ for some $[a,b]\subset\mathbb{R}$. When $z\in\mathbb{R}$ it is now hard to see that $g_z$ is a metric tensor yielding the familiar Laplace-Beltrami operator $\Delta_g$. An article I am reading states that 1.) the family of metrics $g_z$ is analytic in a complex neighbourhood of $[a,b]$ and 2.) the $g_z$s yield some holomorphic family of elliptic operators $L_z$, which for $z$ real correspond to the Laplace-Beltrami operator (henceforth abbreviated as L-B) $\Delta_{g_z}$. Unfortunately I do not know much about elliptic operators nor about analyticity of Riemannian metrics so it is hard for me alone to verify this statement, but knowing that a Riemannian metric $g$ determines the L-B operator $\Delta_g$ I am left to believe that the author is saying that the "some" elliptic operators $L_z$ have a representation like that of the L-B, but the $g_z$ in question can contain complex coefficients. Therefore my questions (which might be best answered with literature references if you happen to know any)
1.) What does it mean for a such $g_z$ to be analytic?
2.) Is my claim regarding the representation of the "some" $L_z$s correct?
Thanks!