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Required fields*

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    $\begingroup$ Something is not quite right here - you would claim that Willmore energy is not only Möbius invariant but actually conformally invariant? But that is not true - consider two different conformal immersions of the sphere. They will not in general have the same Willmore energy, but one is certainly a conformal transformation of the other since there is only one conformal structure on $S^2$. $\endgroup$ Commented Jun 14, 2012 at 16:07
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    $\begingroup$ @fuzzytron, conformal transformations of $\mathbb R^3.$ These come from stereographic projection to $\mathbb S^3,$ some isometry there, then stereographic projection back. $\endgroup$ Commented Jun 15, 2012 at 3:46
  • $\begingroup$ @Will, fuzzytron. It is even true for conformal changes of the metric, not only Moebius transformations, as was shown in "Weiner, J., (1978) On a problem of Chen, Willmore, et al., Indiana University Mathematical Journal." $\endgroup$ Commented Jun 15, 2012 at 20:20
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    $\begingroup$ The Willmore energy is not invariant under inversions centred at some point on the image -- this is well-known. Some energy is lost depending on density. Similarly, images which stretch off to infinity gain energy under inversion. $\endgroup$ Commented Jun 16, 2012 at 22:08