Seminar Analysis: Ruben Jakob (University of Tübingen)

Sufficient conditions for Willmore-immersions in R^3 to be minimal surfaces

We provide two sharp sufficient conditions for immersed Willmore surfaces in R³, definedon bounded C⁴-subdomains of R², to be already minimal surfaces, i.e. to have vanishing mean curvatures on their entire domains. Our precise results read as follows:

Theorem 1. For some bounded C⁴-domain Ω⊂R² let X∈C⁴(Ω,R³) denote some immersed Willmore surface with Gauss map N and mean curvature H. Furthermore, assume that there exist constants c,d∈R and some fixed vector V∈S² such that χ := cX+dV satisfies at least one of the following two conditions:

a) There is some “normal domain” G⊂Ω such that there hold H=0 on ∂G and H≥0 (or H≤0) in G∩O, where O⊂R² is some open neighbourhood of ∂G, and

inf_∂G<χ,N> ≥ 0 as well as sup_∂G<χ,N> > 0;

b) H=0 on ∂Ω and

<χ,N> > 0 in Ω\A as well a sup_∂Ω<χ,N> > 0

for some finite set A⊂Ω.

Then H≡0 is satisfied in \bar{Ω}, i.e.X is a minimal surface on \bar{Ω}.

These results turn out to be particularly suitable for applications to Willmore graphs. We can therefore show that Willmore graphs on bounded C4 domains \bar{Ω} with vanishing mean curvatures on the boundary ∂Ω must already be minimal graphs. Our methods also prove that any closed Willmore surface in R3 which can be represented as a smooth graph over S2 has to be of constant, non-zero mean curvature and therefore 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 R3.

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