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This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...

Let $M$ be a closed manifold, I consider first order laplacian perturbation associated to a density $\rho \in \mathcal{C}^\infty(M)$ with $\rho > 0$ of the form : $$ \Delta_{\rho} f = \Delta f + \...