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Some results about the existence of critical points for the Willmore functional. (English) Zbl 1205.53046
The author proves existence and multiplicity of critical points for the Willmore functional in an ambient manifold \(({\mathbb R}^3, g_\varepsilon)\), where \(g_\varepsilon\) is a metric close and asymptotic to the Euclidean metric. More precisely, let \(g_\varepsilon = g_0 + \varepsilon h\) with \(\lim_{|p|\to \infty}|h_{ij}(p)| = 0\), where \(g_0\) is the Euclidean metric and \(h\) is a symmetric smooth bilinear form. In this case, the scalar curvature of \(({\mathbb R}^3, g_\varepsilon)\) can be written as \(S_{g_\varepsilon} = \varepsilon S_1 + o(\varepsilon)\) where \(S_1 = \sum_{i,j}D^2_{ij}h_{ij} - \Delta {\text{tr}}h.\) Recall that, given an immersed compact oriented surface \(N \hookrightarrow {\mathbb R}^3\), the Willmore functional is defined by \({\mathcal W}(N) = \int_{N} H^2\, d\Sigma,\) where \(H\) and \(d\Sigma\) are the mean curvature and the area form of \(N\), respectively. Under these circumstances, the author proves that, if there is a point \(p \in {\mathbb R}^3\) such that \(S_1(p) \neq 0\) and there exist \(C>0\) and \(\alpha >2\) such that \(|D_k h_{ij}(p)| < \frac{C}{|p|^\alpha}\) for all \(i,j,k =1,2,3\), then, for \(\varepsilon\) small enough, there exist \((p_\varepsilon, r_\varepsilon) \in {\mathbb R}^3 \oplus {\mathbb R}^+\) and \(w_\varepsilon \in C^{4, \alpha}(S^2)\) with \(\| w_\varepsilon\|_{C^{4, \alpha}} \to 0\) as \(\varepsilon \to 0\), such that the perturbed sphere \(S_{r_\varepsilon}(p_\varepsilon, w_\varepsilon)\) is a critical point of the Willmore functional \(\mathcal W\). Here, \(S^2\) is the standard unit sphere and \(S_r(p)\) the standard sphere of \({\mathbb R}^3\) with center \(p\) and radius \(r\) parametrized by \(\theta \to p + r\theta\), and \(S_r(p, w)\) is a perturbed sphere defined as the image of \(\theta \to p + r(1-w(\theta))\theta\) for a given small function \(w\in C^{4, \alpha}(S^2)\).
Furthermore, under the same assumptions, if there exist two points \(p_1, p_2\in {\mathbb R}^3\) such that \(S_1(p_1) >0\) and \(S_1(p_2) < 0\), then there exist two distinct perturbed spheres which are critical points of the Willmore functional. The author also shows a gap theorem stating a non-existence result. Namely, if the scalar curvature is not zero at some point \(\bar p\), then there exist \(r_0 >0\) and \(\rho>0\) such that, for a radius \(r < r_0\) and a perturbation \(w\in C^{4, \alpha}(S^2)\) with \(\| w\|_{C^{4, \alpha}(S^2)} < \rho\), the surfaces \(S_r(p, w)\) are not critical points of the Willmore functional.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E99 Variational problems in infinite-dimensional spaces
35J60 Nonlinear elliptic equations
53A05 Surfaces in Euclidean and related spaces
53C43 Differential geometric aspects of harmonic maps
53C40 Global submanifolds
49Q10 Optimization of shapes other than minimal surfaces
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