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Let $(M,\mu)$ be a compact Riemannian manifold of dimension $n$ and $\theta$ a differential 1-form. Write $\theta=dg+\delta\mu+h$, 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?

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?

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

Let $(M,\mu)$ be a compact Riemannian manifold of dimension $n$ and $\theta$ a differential 1-form. Write $\theta=dg+\delta\mu+h$, 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?