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phase border in a quite large range of the capillary num- ber. 

We hope that this work will incite future experimental research. It will also be interesting to investigate ex- perimentally the presence of widening of the VB band as a function of shear rate. Finally, we did not in- clude third order spherical harmonics in the expansion of the shape function, restricting our calculations only to centro-symmetric vesicles. In the full numerical simu- lation [16], no symmetry restriction is imposed and so far no manifestation of the third order harmonic has been observed. This confers to our assumption of centro- symmetry a certain legitimacy. 

Acknowledgements: We would like to thank G. Danker, S. S. Vergeles, and P. M. Vlahovska for helpful discussions. A.F. and C.M. acknowledge financial sup- port from CNES (Centre National D’études Spatiales) and ANR (Agence Nationale pour la Recherche); "MOSI- COB project". 

Appendix A: The shape evolution equations and the expression of various coefficients in terms of physical parameters 

We remind that 

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The coefficients are then written as 

<img src='content_image/423508.jpg'>

<img src='content_image/423511.jpg'>

