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Energy quantization for Willmore surfaces and applications. (English) Zbl 1325.53014

The principal aim of the paper under review is to understand the closure of the space of Willmore immersions with Willmore energy below a certain level. The authors consider immersions \(\phi:\Sigma\to E^m\) of a \(2\)-dimensional closed domain manifold into the Euclidean \(m\)-space of dimension \(m\geq 3\) that are critical for the energy functional \(W(\Phi)=\int_\Sigma|H|^2dA\), where \(H\) denotes the mean curvature vector field and \(dA\) the induced area element of \(\phi\).
To this end the authors study the sequences of Willmore immersions. Assuming that the corresponding sequence of induced conformal structures on the domain surface stays in a compact subdomain of the moduli space, they identify three regions in the domain surface so that, after possibly passing to a subsequence, convergence can be controlled: the “main” region, where strong convergence holds; one of concentrating parametrization of non-trivial Willmore spheres; and the “bubble and neck regions” connecting the two former ones, where the energy is shown to vanish.
As a consequence the authors derive strong compactness in the \(C^l\)-topology, \(l\in{\mathbb N}\), of the space of Willmore immersions below a suitable energy level. This generalizes results for Willmore tori, \(\Sigma=T^2\), in dimensions \(m=3\) by E. Kuwert and R. Schätzle [Ann. Math. (2) 160, No. 1, 315–357 (2004; Zbl 1078.53007)] and \(m=4\) by the second author [Invent. Math. 174, No. 1, 1–45 (2008; Zbl 1155.53031)].

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A30 Conformal differential geometry (MSC2010)
49Q10 Optimization of shapes other than minimal surfaces
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
58D10 Spaces of embeddings and immersions
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