Good day to everyone! Does anybody know if there are *upper bound* estimates for Willmore energy for a given surface?



Upper bound for Willmore energy
Of course, there are many surfaces for which the Willmore energy can be computed explicitely, for example the Clifford torus and for all Willmore spheres.


An important class of surfaces where one gets upper bounds for the Willmore energy are the Lawson surfaces $\xi\_{k,l}$: for $k\geq l$ one has
$$W( \xi\_{k,l})< 4\pi (l+1),$$
see Kusner "Comparison surfaces for the Willmore problem" in 
Pacific J. Math. 138 (1989), no. 2, 317–345, and the references therein.
In particular, this proves that for every compact, orientable (topological) surface there is an embedding with Willmore energy below $8\pi$ and hence (by Li-Yau) a Willmore minimizer in the respective topological class by a result of L. Simon (Existence of surfaces minimizing the Willmore functional).


