Good day to everyone! Does anybody know if there are upper bound estimates for Willmore energy for a given surface?
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1$\begingroup$ It suffices to compute or estimate the Willmore energy for an explicit embedding of the surface. I imagine there are a number of ways to do this. One possible approach for a surface of genus $g$ is to embed $g$ copies of the torus explicitly and glue them together by removing small disks and attaching a small cylinder that flattens out at both ends. The total Willmore functional of the surface is roughly equal to the sum of the Willmore functionals of the $g$ tori and $g-1$ cylinders. These can be estimated using the explicit formulas defining them. $\endgroup$– Deane YangCommented Jan 20, 2016 at 3:53
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2 Answers
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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).
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For the Otsuki torus $O_{p/q}$ ($p/q$ rational number with $p,q>0$, $(p,q)=1$, $1/2<p/q<1/\sqrt 2$) the Willmore energy $W$ is bounded by $$4\pi q< W < \sqrt{2}\pi^2 q,$$ see On Otsuki tori and their Willmore energy.
answered Jan 19, 2016 at 17:45