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Let $(M^2,g)$ and $(\bar{M},\bar{g})$ be two Riemannian manifolds. Suppose that $\mathcal{W}$ is the Willmore functional on the set of immersion functions from $M^2$ to $\bar{M}$.

We know that $(\bar{M},\bar{g})$ is supposed to be of constant sectional curvature at every paper where the critical points of Willmore functional have been investigated. My question is: may I suppose $(\bar{M},\bar{g})$ to be any Riemannian manifold?

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    $\begingroup$ If $\bar{M}$ is not 3-dimensional, I suppose you have to say what you mean by Willmore energy. There is more than one candidate, as far as I can see. But there is no problem if $\bar{M}$ is 3-dimensional, integral of squared mean minus integral of Gauss. $\endgroup$
    – Ben McKay
    Apr 7, 2016 at 9:21

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