Let $F:M\to\overline M$ be an immersion of a manifold $M$ into a Riemannian manifold $(\overline M, \overline g)$ and let $H\in C^\infty(M)$ be a given function. I would like to find a functional $I_H(F)$ whose stationary points are the immersions whose mean curvature is the given $H$, namely $$ I_H'(0) = 0 \ \text{along any compactly supported variation} \iff H_F=H $$ where $H_F$ is the mean curvature of $F$. It should be a functional of the form $$ I_H(F)=A(F)+J_H(F)=\int_M dx_g + J_H(F)$$ where $g=F^*\bar g$ is the induced metric and $dx_g$ is its Riemannian volume. Does anyone know who is $J_H$?
I know that in the case of graphs this functional is given by $$ I_H(u)=\int_{\mathbb{R}^m}\sqrt{1+|Du|^2} dx-\int_{\mathbb{R}^m} Hu \ dx $$ (here $F:\mathbb{R}^m\to \mathbb{R}^{m+1}: x \mapsto (u(x),x)$) but I can't figure it out in a general context. Thank you.