# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Ambrosetti A., Badiale M.: Homoclinics: Poincaré–Melnikov type results via a variational approach. Ann. Inst. Henri Poincaré Anal. Non Linèaire 15, 233–252 (1998) · Zbl 1004.37043 · doi:10.1016/S0294-1449(97)89300-6  Ambrosetti A., Badiale M.: Variational Perturbative methods and bifurcation of bound states from the essential spectrum. Proc. R. Soc. Edinb. 18, 1131–1161 (1998) · Zbl 0928.34029  Ambrosetti A., Malchiodi A.: Perturbation methods and semilinear elliptic problems in $${\mathbb{R}^n}$$ . Progress in mathematics. Birkhauser, Basel (2006) · Zbl 1115.35004  Ambrosetti A., Malchiodi A., Ni W.-Mi.: Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I. Comm. Math. Phys. 235, 427–466 (2004) · Zbl 1072.35019 · doi:10.1007/s00220-003-0811-y  Ambrosetti A., Malchiodi A.: A multiplicity result for the Yamabe problem on S n . J. Funct. Anal. 168, 529–561 (1999) · Zbl 0949.53028 · doi:10.1006/jfan.1999.3458  Bauer M., Kuwert E.: Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not. 10, 553–576 (2003) · Zbl 1029.53073 · doi:10.1155/S1073792803208072  Blaschke W.: Vorlesungen über Differentialgeometrie, III. Springer, Berlin (1929) · JFM 55.0422.01  Brendle S.: Blow-up phenomena for the Yamabe equation. J. Am. Math. Soc. 21, 951–979 (2007) · Zbl 1206.53041 · doi:10.1090/S0894-0347-07-00575-9  Bressan A.: Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem. Oxford University Press, Oxford (2000) · Zbl 0997.35002  Friesecke G., James R., Müller S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55(11), 1461–1506 (2002) · Zbl 1021.74024 · doi:10.1002/cpa.10048  Guo Z.: Generalized Willmore functionals and related variational problems. Differ. Geom. Appl. 25, 543–551 (2007) · Zbl 1138.53016 · doi:10.1016/j.difgeo.2007.06.004  Jost J.: Partial Differential Equations, Graduate Texts in Mathematics, vol. 214. Springer, Berlin (2002) · Zbl 1034.35001  Kuwert E., Schätzle R.: The Willmore flow with small initial energy. J. Differ. Geom. 57(3), 409–441 (2001) · Zbl 1035.53092  Kuwert E., Schätzle R.: Gradient flow for the Willmore functional. Comm. Anal. Geom. 10(2), 307–339 (2002) · Zbl 1029.53082  Parthasarathy R., Viswanathan K.S.: Geometric properties of QCD string from Willmore functional. J. Geom. Phys. 38(3–4), 207–216 (2001) · Zbl 0987.81116 · doi:10.1016/S0393-0440(00)00062-0  Pacard, F., Xu, X.: Constant mean curvature spheres in Riemannian manifolds. Manuscr. Math. (2008, in press) · Zbl 1165.53038  Rivière T.: Analysis aspects of Willmore surfaces. Invent. Math. 174(1), 1–45 (2008) · Zbl 1155.53031 · doi:10.1007/s00222-008-0129-7  Simonett G.: The Willmore flow near spheres. Differ. Integral Equ. 14(8), 1005–1014 (2001) · Zbl 1161.35429  Simon L.: Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1(2), 281–325 (1993) · Zbl 0848.58012  Thomsen, G.: Über Konforme Geometrie, I: Grundlagen der Konformen Flächentheorie. Abn. Math. Sem. Hamburg 31–56 (1923) · JFM 49.0530.02  White J.H.: A global invariant of conformal mappings in space. Proc. Am. Math. Soc. 38, 162–164 (1973) · Zbl 0256.53008 · doi:10.1090/S0002-9939-1973-0324603-1  Willmore T.J.: Riemannian Geometry. Oxford Science Publications, Oxford University Press, Oxford (1993) · Zbl 0797.53002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.