Given two metrics $g,$ and $h$, related by $h = e^{\frac{2f}{n}}g$, what is the relation between $\star_g$ and $\star_g$?

Question

Given two metrics $g,$ $h$, and a smooth function $f$ on a riemannian manifold $M^n$, supposing that $g$ and $h$ are related by $h = e^{\frac{2f}{n}}g$, what is the relation between $\star_g$ and $\star_h$? Where $\star$ is the Hodge star operator?

Answer 1

That can be found here, it refers to the book on Einstein manifolds. Under the conformal change $\tilde g = e^{2f} g$, it is of the form (when acts on $p$-forms)

$$*_{\tilde g} = e^{(n-2p)f} * _g.$$

For your interest, the codifferential $\delta$ is related by

$$\tilde \delta = e^{-2f} \left( \delta - (n-2p) \iota_{\nabla f}\right).$$

So if $2p = n$, both the Hodge star and the codifferential do not depends on $\nabla f$, which is quite amazing. In $n=2$ (the case of Riemann surfaces), the Hodge star operator on one form can be described solely by the complex/conformal structure of $M$.