Some results about the existence of critical points for the Willmore functional. (English) Zbl 1205.53046

Mondino, Andrea

Math. Z. 266, No. 3, 583-622 (2010).

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.

Reviewer: Gabjin Yun (Yongin)

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

Keywords:

mean curvature; nonlinear elliptic PDE; perturbed method; Willmore functional; scalar curvature

References:

[1]	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
[2]	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
[3]	Ambrosetti A., Malchiodi A.: Perturbation methods and semilinear elliptic problems in \({\mathbb{R}^n}\) . Progress in mathematics. Birkhauser, Basel (2006) · Zbl 1115.35004
[4]	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
[5]	Ambrosetti A., Malchiodi A.: A multiplicity result for the Yamabe problem on S n . J. Funct. Anal. 168, 529–561 (1999) · Zbl 0949.53028
[6]	Bauer M., Kuwert E.: Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not. 10, 553–576 (2003) · Zbl 1029.53073
[7]	Blaschke W.: Vorlesungen über Differentialgeometrie, III. Springer, Berlin (1929) · JFM 55.0422.01
[8]	Brendle S.: Blow-up phenomena for the Yamabe equation. J. Am. Math. Soc. 21, 951–979 (2007) · Zbl 1206.53041
[9]	Bressan A.: Hyperbolic Systems of Conservation Laws, The One-Dimensional Cauchy Problem. Oxford University Press, Oxford (2000) · Zbl 0997.35002
[10]	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
[11]	Guo Z.: Generalized Willmore functionals and related variational problems. Differ. Geom. Appl. 25, 543–551 (2007) · Zbl 1138.53016
[12]	Jost J.: Partial Differential Equations, Graduate Texts in Mathematics, vol. 214. Springer, Berlin (2002) · Zbl 1034.35001
[13]	Kuwert E., Schätzle R.: The Willmore flow with small initial energy. J. Differ. Geom. 57(3), 409–441 (2001) · Zbl 1035.53092
[14]	Kuwert E., Schätzle R.: Gradient flow for the Willmore functional. Comm. Anal. Geom. 10(2), 307–339 (2002) · Zbl 1029.53082
[15]	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
[16]	Pacard, F., Xu, X.: Constant mean curvature spheres in Riemannian manifolds. Manuscr. Math. (2008, in press) · Zbl 1165.53038
[17]	Rivière T.: Analysis aspects of Willmore surfaces. Invent. Math. 174(1), 1–45 (2008) · Zbl 1155.53031
[18]	Simonett G.: The Willmore flow near spheres. Differ. Integral Equ. 14(8), 1005–1014 (2001) · Zbl 1161.35429
[19]	Simon L.: Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom. 1(2), 281–325 (1993) · Zbl 0848.58012
[20]	Thomsen, G.: Über Konforme Geometrie, I: Grundlagen der Konformen Flächentheorie. Abn. Math. Sem. Hamburg 31–56 (1923) · JFM 49.0530.02
[21]	White J.H.: A global invariant of conformal mappings in space. Proc. Am. Math. Soc. 38, 162–164 (1973) · Zbl 0256.53008
[22]	Willmore T.J.: Riemannian Geometry. Oxford Science Publications, Oxford University Press, Oxford (1993) · Zbl 0797.53002