Willmore Surfaces of Revolution with Two Prescribed Boundary Circles

Boundary value problems for Willmore curves in $$\mathbb {R}^2$$ R 2

“…The Navier boundary value problem (2) does not appear to be easily solvable since the curvature function det(γ , γ )|γ | −3 of a given smooth regular Willmore curve γ does in general not solve a reasonable differential equation. If, however, we parametrize γ by arclength we obtain a smooth regular Willmore curveγ : [0, L] → R 2 whose curvature function κ satisfies the differential equation −2κ =κ 3 on [0, L] that is explicitly solvable.…”

“…Then a curve γ solves the Navier problem (2) if and only if γ is given by (9) for admissible parameters a, b, L and Q ∈ SO(2) satisfying (10).…”

“…The curvature functions of such curves satisfy −κ +κ = 1 2κ 3 instead of (7) so that it is tempting to conjecture that boundary value problems for Willmore surfaces of revolution can be solved with a similar analysis as ours. We refer the interested reader to [2][3][4][5]8] for more information on recent advances concerning Willmore surfaces of revolution.…”

“…Next, in Sect. 3, we provide all information concerning the Navier problem starting with a derivation of the complete solution theory for the general Navier problem (2) in Sect. 3.1.…”

“…for given A, B ∈ R 2 and κ 1 , κ 2 ∈ R. In the special case κ 1 = κ 2 =: κ the boundary value problem (2) was investigated by Deckelnick and Grunau [6] under the restrictive assumption that γ is a smooth symmetric graph lying on one side of the straight line joining A and B. More precisely they assumed A = (0, 0), B = (1, 0) as well as γ (t) = (t, u(t)) (t ∈ [0, 1]) for some positive symmetric function u ∈ C 4 (0, 1) ∩ C 2 [0, 1].…”