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I apologize if this question is silly or confusing, I am completely new to this subject.

Let $(M,g)$ be a Riemannian manifold. Denote by $\nabla$ the Levi-Civita connection of $(M,g)$. Now, let $S^{n}$ be an oriented compact manifold. Given an immersion $\phi:S\to M$ , denote by $\phi^{*}\nabla$ the Levi-Civita connection of $\left(S,\phi^{*}g\right) $ , by $ h^{\phi}$ the second fundamental form of $\phi$, and by $H^{\phi}$ the mean curvature vector field. Denote by $\nu_{\phi}$ the only Riemannian volume form on $\left(S,\phi^{*}g\right)$ which induces the given orientation on $S$ . The total tension of the immersion $\phi$ is the integral $\int_{S}\left|\left|H^{\phi}\right|\right|^{2}\nu_{\phi}$. Now, denote by $Imm\left(S,M\right)$ the space of smooth immersions of $S$ into $M$ . Then we can define the total tension functional $E:Imm\left(S,M\right)\to\mathbb{R}$ given by $E\left(\phi\right)=\int_{S}\left|\left|H^{\phi}\right|\right|^{2}\nu_{\phi}$ .

Denote $I=Imm(S,M)$. Then $I$ is a smooth infinite dimensional Frechet space, and its tangent space $T_\phi I$ is the space of smooth vector fields $X:S\to TM$ along $\phi$. We can define in a natural way a Riemannian metric $G$ on $I$ induced by $g$ and the orientation of $N$. It is defined pointwise as $G_\phi (X,Y)=\int_S g(X,Y)\nu_\phi $. This allows us to consider the flow of the vector field $-grad E$.

I am sure this flow has been studied extensively. Can you point me some reference to articles about it? Thank you.

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    $\begingroup$ It seems to be the Willmore flow if $n=2$ and when $M$ is euclidean. $\endgroup$ – Arctic Char Nov 23 '16 at 8:55
  • $\begingroup$ @JohnMa yes it is. However, in general the Willmore functional for general immersions is different from the one I gave. $\endgroup$ – Ervin Nov 23 '16 at 15:50

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