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Existence of surfaces minimizing the Willmore functional. (English) Zbl 0848.58012
The Willmore functional is defined for compact surfaces \(\Sigma\) embedded in \(\mathbb{R}^n\) by \({\mathcal F}(\Sigma) = 4^{-1} \cdot \int_\Sigma |H|^2\), where the integration is with respect to ordinary 2-dimensional area measure, and \(H\) is the mean curvature vector of \(\Sigma\). The existence problem of a surface of genus \(g\) minimizing \(\mathcal F\) is studied. It is proved that the minimizing surface exists in the case \(g = 1\). For \(g\) greater than 1 it is shown that there is an embedded real analytic surface \(\Sigma_g\) of genus \(g\) in \(\mathbb{R}^n\) minimizing \(\mathcal F\) unless for some choice of \(q \geq 2\), \(\ell_1, \ell_2, \dots, \ell_q\), \(\sum^q_{j =1} \ell_j = g\), the equality \(\ell_g = \sum^q_{j =1} \ell_j\) holds, where \(\ell_g = \inf {\mathcal F} (\Sigma_g) -4\pi\). The inequality \(\ell_g \leq \sum^q_{j = 1} \ell_j\) is proved to be valid for any integers \(q \geq 2\) and \(\ell_1,\dots, \ell_q \geq 1\) with \(\sum^q_{j = 1} \ell_j = g\). It is conjectured that the strict inequality always takes place.

MSC:
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
49Q05 Minimal surfaces and optimization
53A05 Surfaces in Euclidean and related spaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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