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Let $(M,\mu)$ be a compact Riemannian manifold of dimension $n$ and $\theta$ a differential 1-form. Write $\theta=dg+\delta\phi+h$, the decomposition of $\theta$ according to the Hodge decomposition. I came across the following equation $$\Delta(f-g)=\frac{n-2}{2(n-1)}\mu(df,df-\theta)$$where $f$ is the unknown and $\Delta$ is the Laplacian associated to the metric.

Is there some general result which ensures the existence of a solution of this equation?

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  • $\begingroup$ A bit of a long shot, but have you tried to consider the map sending $f_0$ to the solution of $\Delta(f-g) = F(f_0)$, where $F$ is the right-hand side of your equation (which by the way I cannot interpret, what are those two 1-form as arguments of a function?) With a (good) bit of luck, you could have the contraction property in a suitable norm, providing existence and uniqueness of the solution. $\endgroup$
    – Benoît Kloeckner
    Commented Mar 25, 2021 at 10:46
  • $\begingroup$ You are well come to MO! $\endgroup$
    – Ali Taghavi
    Commented Mar 25, 2021 at 19:36
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    $\begingroup$ Am I mistaken to think that there is a possible confusion for 2-form $\mu$ in the second line and the initial metric $\mu$? $\endgroup$
    – Ali Taghavi
    Commented Mar 25, 2021 at 19:54
  • $\begingroup$ You are right. I fixed. $\endgroup$
    – Mohamed Boucetta
    Commented Mar 26, 2021 at 13:54

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