Sufficient conditions for Willmore immersions in $$\mathbb{R }^3$$ R 3 to be minimal surfaces

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 \(\mathbb{R }^3\).