Monotonicity of infimum of the Willmore energy with prescribed genus

As pointed out in the paper you reference, this question was posed on p. 446 of this paper

Kühnel, Wolfgang; Pinkall, Ulrich, On total mean curvatures, Q. J. Math., Oxf. II. Ser. 37, 437-447 (1986). ZBL0627.53044.

I found 19 papers citing this paper on Google scholar. Of these, one is a survey paper by Kuwert and Schätzle from 2012 which reproves the Kuwert-Li-Schäzle result (and doesn't mention the monotonicity question). A cursory look at the titles and abstracts indicates that this question doesn't seem to have been resolved (although there are several papers by Kuwert-Schäzle which address related questions). Presumably a paper proving monotonicity would cite their paper, so I'm guessing that this is still an open question since 1986.

Tags:

Dg.Differential Geometry

Related

Are infinite groups in which most elements have order $\leq 2$ commutative? Are mapping class groups of orientable surfaces good in the sense of Serre? On the Fourier-Laplace transform of compactly supported distributions Localization of $\infty$-categories Groups whose poset of direct factors are lattices Does $M^o=N^o$ imply that $\partial M = \partial N$? Residues of $\frac{1}{\prod_{i=1}^n (x-P_i)^{e_i}}$ Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise Does Koszul duality between $Comm$ and $Lie$ imply the power series identity $\exp(\ln(1-z))-1 = -z$? Has it been proved that weak solutions to the Navier-Stokes equations are non-unique, and does this prove that the Navier-Stokes are not valid? Reference on Persistent Homology Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?

Recent Posts