Willmore minimizers for genus $\geq 2$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T18:16:08Z http://mathoverflow.net/feeds/question/92013 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92013/willmore-minimizers-for-genus-geq-2 Willmore minimizers for genus $\geq 2$ jo1o 2012-03-23T14:43:39Z 2012-11-29T16:19:25Z <p>For an immersed closed surface $f: \Sigma \rightarrow \mathbb R^3$ the Willmore functional is defined as $$\cal W(f) = \int _{\Sigma} \frac{1}{4} |\vec H|^2 d \mu_g,$$ where $\vec H$ is the mean curvature vector in $\mathbb R^3$and $g$ is the induced metric.</p> <p>If $\Sigma$ is closed we have the estimate $$\cal W(f) \geq 4 \pi$$ with equality only for $f$ parametrizing a round sphere. </p> <p>Recently, the Willmore conjecture was proved (the paper can be found on arxiv), which states that for closed surfaces $\Sigma$ of genus $g \geq 1$ this estimate can be improved: $$\cal W(f) \geq 2 \pi^2$$ with equality only for the Cilfford torus. </p> <p>Are there any conjectures about the minimizers in the case of genus $g \geq 2$? And what happens if we consider surfaces immersed in some $\mathbb R^n$ instead of $\mathbb R ^3$?</p> http://mathoverflow.net/questions/92013/willmore-minimizers-for-genus-geq-2/92014#92014 Answer by Sebastian for Willmore minimizers for genus $\geq 2$ Sebastian 2012-03-23T14:54:27Z 2012-03-23T14:54:27Z <p>First of all, by a result of Bauer and Kuwert, there exists a smooth minimizer of the Willmore functional in the class of compact surfaces with fixed genus g, for any g. They have Willmore functional below $8\pi$ and by a result of Kuwert, Li and Schaetzle, the Willmore functional of the minimzers for genus $g$ tends to $8\pi$ when $g$ goes to infinity. Not much more is known about higher genus surfaces, but there is a vague conjecture, that the minimzers are the so called Lawson surface $\xi_{g,1}.$</p> http://mathoverflow.net/questions/92013/willmore-minimizers-for-genus-geq-2/114899#114899 Answer by Yong Luo for Willmore minimizers for genus $\geq 2$ Yong Luo 2012-11-29T16:19:25Z 2012-11-29T16:19:25Z <p>I remember there is a paper by Kusner named: comparison surfaces for the Willmore problem in which the author conjectured that the Lawson surface(see Sebastian's answer) minimizes the Willmore energy of genus g surface. For surfaces immersed in R^n, it is also conjectured the Clifford torus should be the minimizer, but it seems to me that this is still an open question.</p>