Existence of surfaces minimizing the Willmore functional

For compact surfaces E embedded in R", the Willmore functional is defined by ^) = i/jHr where the integration is with respect to ordinary 2-dimensional area measure, and H is the mean curvature vector of S (in case n = 3 we have |H| = Ifti + A^l?where «i, K2 are the principal curvatures of E).In particular ^"(S 2 ) = 47r.For surfaces E without boundary we have the important fact that ^"(E) is invariant under conformal transformations of M n ; thus if E C M n is the image of E under an isometry or a scaling (x i-> Ax, A > 0) or an inversion in a sphere with centre not in E (e.g.x >-> x/|a:| 2 if 0 ^ E) then (0.1) ^(E)=^(S).(See [WJ], [LY], [W] for general discussion.)For each genus g = 0,1,2,... and each n > 3 we letwhere the inf is taken over all compact genus g surfaces without boundary embedded in R n .We note some inequalities concerning the numbers /?™.Firstly we claim (0.2) 47r < /?" < STT