Consider a Riemannian manifold $(M,g_0)$ which is the interior of a compact manifold $(\overline{M}, \overline{g})$.
I'm interested in a kind of conformal variation of the background metric $g_0$. So for $\tau \in [0,T] \subset \mathbb{R}$ let $\omega_\tau \in C^{\infty}(\overline{M})$ and $g_{\tau} := e^{\omega(\tau)} g_0$
In addition there is a family of operators $\{H_\tau \}$ where $H_\tau: L^2(M, g_\tau) \rightarrow L^2(M, g_\tau)$. My goal is to talk about the $(L²-)$ trace of these operators with respect to the background $L^2$-space $L^2(M,g_0)$.
Is this possible? Are there suitable conditions for $\omega$ to get a statement like \begin{equation} L^2(M,g_0)= L^2(M, g_\tau) \ \forall \tau \in [0,T] \end{equation} so that the operators can be viewed as acting on the same space?