Abstract
We provide two sharp sufficient conditions for immersed Willmore surfaces in \(\mathbb{R }^3\) to be already minimal surfaces, i.e. to have vanishing mean curvature on their entire domains. These results turn out to be particularly suitable for applications to Willmore graphs. We can therefore show that Willmore graphs on bounded \(C^4\)-domains \(\overline{\varOmega }\) with vanishing mean curvature on the boundary \(\partial \varOmega \) must already be minimal graphs, which in particular yields some Bernstein-type result for Willmore graphs on \(\mathbb{R }^2\). Our methods also prove the non-existence of Willmore graphs on bounded \(C^4\)-domains \(\overline{\varOmega }\) with mean curvature \(H\) satisfying \(H \ge c_0>0 \,{\text{ on }}\, \partial \varOmega \) if \(\varOmega \) contains some closed disc of radius \(\frac{1}{c_0} \in (0,\infty )\), and they yield that any closed Willmore surface in \(\mathbb{R }^3\) which can be represented as a smooth graph over \(\mathbb{S }^2\) has to be a round sphere. Finally, we demonstrate that our results are sharp by means of an examination of some certain part of the Clifford torus in \(\mathbb{R }^3\).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.