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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)

##### MSC:
 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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