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Existence of minimizing Willmore surfaces of prescribed genus. (English) Zbl 1029.53073
The authors consider an immersed surface \(f:\Sigma \to\mathbb{R}^n\), the Willmore integral \({\mathcal W}(f)= \int_\Sigma|{\mathcal H}|^2 d\mu\), \(({\mathcal H}=\) the mean curvature vector and \(\mu=\) the induced area measure) and \(\beta_{\mathfrak p}=\inf \{{\mathcal W}(f); f\in{\mathcal S}_{\mathfrak p}\}\) \(({\mathcal S}_{\mathfrak p}=\) the class of immersions \(f\), where \(\Sigma\) is an orientable, closed surface with genus \((\Sigma)={\mathfrak p})\). They prove (Theorem 1.2): For any \({\mathfrak p}\in \mathbb{N}_0\) the infimum \(\beta_{\mathfrak p}\) is attained by an oriented, closed Willmore surface of genus \({\mathfrak p}\).
Reviewer: A.Neagu (Iaşi)

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C43 Differential geometric aspects of harmonic maps
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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