0
$\begingroup$

Let us assume $\text{H}$ is the mean curvature of a compact surface in $\mathbb E^3$ and $g$ is its genus.

  • When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and $\int_{\Sigma}\text{H}^2dV\geq4\pi$.

  • When $g=1$, we have $\int_{\mathbb T^2}\text{H}^2dV\geq2\pi^{2}$ and $\int_{\Sigma}\text{H}^2dV\geq2\pi^{2}$.

  • I want to ask that these theorems can be generalized to a compact surface with $g>1$ in $\mathbb E^3$? And the statement below is right or not?
    $$\text{When}~~g(\Sigma_{1})>g(\Sigma_{2})\text{, then do we have} \int_{\Sigma_{1}}\text{H}^2dV\geq\int_{\Sigma_{2}}\text{H}^2dV\text{?}$$

$\endgroup$
3
  • 5
    $\begingroup$ Look up the literature on the Willmore problem, and you will get the answers to your questions. A good place to start is en.wikipedia.org/wiki/Willmore_conjecture, for example. $\endgroup$ Dec 24, 2013 at 16:11
  • 3
    $\begingroup$ Google "Willmore functional", and check out this question: mathoverflow.net/questions/92013/… $\endgroup$
    – Igor Rivin
    Dec 24, 2013 at 16:11
  • 1
    $\begingroup$ As for (3), as far as I know, the first person to study this question was Leon Simon: ams.org/mathscinet-getitem?mr=1243525, who proved certain existence results for minimizers. This paper is very beautiful and the techniques in here have been applied repeatedly to other problems. $\endgroup$ Dec 24, 2013 at 17:51

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.