The conformal Willmore functional: a perturbative approach. (English) Zbl 1276.53068

The conformal Willmore functional of a compact orientable surface \(S\) embedded in a three-dimensional Riemannian manifold \((M,g)\) is defined to be the integral \[ I(S) = \int_S \left( {H^2 \over 4} - D \right) d\Sigma, \] where \(H\) is the mean curvature, \(D\) is the product of the principal curvatures, and \(d\Sigma\) is the area form of \(S\) induced from the embedding of \(S\) in \(M\). This functional is conformally invariant which means that if \(\Psi : M \to M\) is a conformal diffeomorphism, then \(I(\Psi(S)) = I(S)\).
The critical points of \(I\) are called conformal Willmore surfaces. In the paper under review, the author investigates existence and nonexistence of conformal Willmore surfaces using a perturbative method called Lyapunov-Schmidt reduction. It is noteworthy that, unlike previous studies of conformal Willmore surfaces, the ambient manifolds considered here do not have constant curvature.
In the case when \((M,g)\) is \(({\mathbb R}^3, g_\varepsilon)\), where \(g_\varepsilon\) is a certain metric sufficiently close to and asymptotic to the flat Euclidean metric on \({\mathbb R}^3\), the author shows that there exist perturbed spheres which are conformal Willmore surfaces. On the other hand, for a general three-dimensional Riemannian manifold \((M,g)\), if the traceless Ricci tensor of \(M\) does not vanish at the point \(p \in M\), then no sufficiently small perturbed sphere about \(p\) can be a Willmore surface.


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C20 Global Riemannian geometry, including pinching
58E99 Variational problems in infinite-dimensional spaces
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35J60 Nonlinear elliptic equations
Full Text: DOI arXiv


[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. Birkhäuser, Basel (2006) · Zbl 1115.35004
[4] Bauer, M., Kuwert, E.: Existence of minimizing Willmore surfaces of prescribed genus. Int. Math. Res. Not. 10, 553–576 (2003) · Zbl 1029.53073
[5] Chen, B.Y.: Some conformal invariants of submanifolds and their applications. Boll. Un. Mat. Ital., Ser. 4 10, 380–385 (1974) · Zbl 0321.53042
[6] Guo, Z.: Generalized Willmore functionals and related variational problems. Differ. Geom. Appl. 25, 543–551 (2007) · Zbl 1138.53016
[7] Guo, Z., Li, H., Wang, C.: The second variational formula for Willmore submanifolds in S n . Results Math. 40(1–4), 205–225 (2001) · Zbl 1163.53312
[8] Hu, Z., Li, H.: Willmore submanifolds in a Riemannian manifold. In: Cont. Geom. Rel. Topics, pp. 251–275 (2004) · Zbl 1076.53071
[9] Kuwert, E., Schätzle, R.: Removability of isolated singularities of Willmore surfaces. Ann. Math. 160(1), 315–357 (2004) · Zbl 1078.53007
[10] Li, H., Vrancken, L.: New examples of Willmore surfaces in S n . Ann. Glob. Anal. Geom. 23(3), 205–225 (2003) · Zbl 1033.53049
[11] Ma, X., Wang, P.: Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms. Sci. China Ser. A 51(9), 9 (2008) 1561–1576 · Zbl 1163.53005
[12] Malchiodi, A.: Existence and multiplicity results for some problems in differential geometry. Ph.D. Thesis, SISSA, October (2000)
[13] Mondino, A.: Some results about the existence of critical points for the Willmore functional. Math. Z. 266(3), 583–622 (2010) · Zbl 1205.53046
[14] Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966) · Zbl 0142.38701
[15] Pacard, F., Xu, X.: Constant mean curvature spheres in Riemannian manifolds. Manuscr. Math. 128(3), 275–295 (2009) · Zbl 1165.53038
[16] Pedit, F.J., Willmore, T.J.: Conformal geometry. Atti Semin. Mat. Fis. Univ. Modena 36(2), 237–245 (1988) · Zbl 0665.53048
[17] Petersen, P.: Riemannian Geometry, 2nd edn. Graduate Texts in Mathematics, vol. 171. Springer, Berlin (2006) · Zbl 1220.53002
[18] Rivière, T.: Analysis aspects of Willmore surfaces. Invent. Math. 174(1), 1–45 (2008) · Zbl 1155.53031
[19] Simon, L.: Existence of surfaces minimizing the Willmore functional. Commun. Anal. Geom. 1(2), 281–325 (1993) · Zbl 0848.58012
[20] Weiner, J.L.: On a problem of Chen, Willmore, et al. Indiana Univ. Math. J. 27(1), 19–35 (1978) · Zbl 0368.53043
[21] Willmore, T.J.: Riemannian Geometry. Oxford Science Publications. Oxford University Press, Oxford (1993) · Zbl 0797.53002
[22] Wei, G.: New examples of Willmore hypersurfaces in a sphere. Houst. J. Math. 35(1), 81–92 (2009) · Zbl 1166.53042
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.