Uniqueness theorems for Willmore surfaces with fixed and free boundaries (Q2733839)

Let \(\Sigma\) be a compact two-dimensional manifold with boundary \(\partial \Sigma\) and \(\Omega\) and \(B\) subsets of \(\mathbb{E}^3\) with \(B\subset \Omega\). Then the author studies \(C^4\) Willmore immersions \(X:\Sigma\to\Omega\) with \(X(\partial\Sigma)\subset B\) (which by definition are critical points of the functional \(X\mapsto \int_\Sigma (H^3_X-K_X) dA)\). At first he shows in the case where \(\Sigma\) is a disc, \(\Omega= \mathbb{E}^3\) and \(B=S^1 \subset P:=\{x_3= 0\}\) that \(X(\Sigma)\) is a spherical cap or a disc, if \(X\) has a constant angle along \(\partial \Sigma\) with the plane \(P\). It is also proved that there exist such ``Willmore discs'' which intersect the plane \(P\) in a non constant angle. In the second part \(B\) is the boundary of \(\Omega\), it means, the author studies the free boundary problem; in particular the case is treated where \(\Omega\) is a half space.