0

I am currently studying about Willmore energy and out of my expectation, I produced a simple result that any surface that is topologically a sphere (genus 0) cannot be a minimal surface. (I am not sure this result appear somewhere or not) Here is my proof:

As we know, the Willmore energy for a surface $S$ that is topologically a sphere is $$W(S)\geq 4\pi^2$$ and Willmore energy is defined as $$W(S)=\int_SH^2dA$$ Suppose that $S$ is a minimal surface. Then $H\equiv 0$ and thus $W(S)=0$, a contradiction. Therefore, there is no surface topologically a sphere is a minimal surface.

Nothing
  • 1,768
  • a minimal surface can not be homeomorphic to $S^2$ since $S^2$ is compact. see https://math.stackexchange.com/questions/544134/there-are-no-compact-minimal-surfaces – Zest Jun 03 '20 at 14:59
  • Hmm, thanks. Then my guess is correct. – Nothing Jun 03 '20 at 15:41

0 Answers0