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The isoperimetric ratio of an embedded surface in $R^3$ is defined as theratio of the area of the surface to power three to the squared enclosed volume.The aim of the present work is to study the minimization of the Willmore energyunder fixed isoperimetric ratio when the underlying abstract surface has fixedgenus $g\geq 0$. The corresponding problem in the case of spherical surfaces,i.e. $g=0$, was recently solved by Schygulla with different methods.

Embedded surfaces of arbitrary genus minimizing the Willmore energy under isoperimetric constraint