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Rivière, Tristan

Analysis aspects of Willmore surfaces. (English) Zbl 1155.53031

Invent. Math. 174, No. 1, 1-45 (2008).
Author’s abstract: In the present paper, a new formulation for the Euler-Lagrange equation of the Willmore functional for immersed surfaces in \(\mathbb R^{m}\) is given as a nonlinear elliptic equation in divergence form, with non-linearities comprising only Jacobians. Letting \(\overrightarrow{H}\) be the mean curvature vector of the surface, the new formulation reads \(\mathcal{L}\overrightarrow{H}=0\), where \(\mathcal{L}\) is a well-defined locally invertible self-adjoint elliptic operator. Several consequences are studied. In particular, the long standing open problem asking for a meaning of the Willmore Euler-Lagrange equation for immersions having only \(L^2\)-bounded second fundamental form is solved. The regularity of weak Willmore immersions with \(L^2\)-bounded second fundamental form is also established. Its proof relies on the discovery of conservation laws which are preserved under weak convergence.
A weak compactness result for Willmore surfaces with energy less than \(8\pi\) (the Li-Yau condition ensuring the surface is embedded) is proved, via a point removability result established for Willmore surfaces in \(\mathbb R^m\), thereby extending to arbitrary codimension the main result in [(*) E. Kuwert and R. Schätzle, Ann. Math. (2) 160, No. 1, 315–357 (2004; Zbl 1078.53007)]. Finally, from this point-removability result, the strong compactness of Willmore tori below the energy level \(8\pi\) is proved both in dimension \(3\) (this had already been settled in [(*)] and in dimension \(4\).
Reviewer: Huafei Sun (Beijing)

Cited in 2 Reviews
Cited in 85 Documents

MSC:

53C43 Differential geometric aspects of harmonic maps
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces

Keywords:

Willmore surface; Euler-Lagrange equation; point removability

Citations:

Zbl 1078.53007
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Full Text: DOI Link

References:

[1] Bernard, Y., Rivière, T.: Palais Smale sequences of the Willmore functional. In preparation (2008)
[2] Blaschke, W.: Vorlesungen über Differential Geometrie III. Springer, Berlin (1929) · JFM 55.0422.01
[3] Bryant, R.L.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20(1), 23–53 (1984) · Zbl 0555.53002
[4] Chen, B.-Y.: Some conformal invariants of submanifolds and their applications. Boll. Unione Mat. Ital. (4) 10, 380–385 (1974) · Zbl 0321.53042
[5] Coifman, R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl., IX. Sér 72(3), 247–286 (1993) · Zbl 0864.42009
[6] Federer, H.: Geometric Measure Theory. Springer, New York (1969) · Zbl 0176.00801
[7] Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55(11), 1461–1506 (2002) · Zbl 1021.74024 · doi:10.1002/cpa.10048
[8] Hawking, S.W.: Gravitational radiation in an expanding universe. J. Math. Phys. 9, 598–604 (1968) · doi:10.1063/1.1664615
[9] Hélein, F.: Harmonic Maps, Conservation Laws, and Moving Frames. Camb. Tracts Math., vol. 150. Cambridge University Press, Cambridge (2002) · Zbl 1010.58010
[10] Helfrich, W.: Elastic properties of lipid bilayers: theory and possible experiments. Z. Nat.forsch. A C28, 693–703 (1973)
[11] Huber, A.: On subharmonic functions and differential geometry in the large. Comment. Math. Helv. 32, 13–72 (1957) · Zbl 0080.15001 · doi:10.1007/BF02564570
[12] Huisken, G., Ilmanen, T.: The Riemannian Penrose inequality. Int. Math. Res. Not. 1997(20), 1045–1058 (1997) · Zbl 0905.53043 · doi:10.1155/S1073792897000664
[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. Commun. Anal. Geom. 10(2), 307–339 (2002) · Zbl 1029.53082
[15] Kuwert, E., Schätzle, R.: Removability of point singularities of Willmore surfaces. Ann. Math. (2) 160(1), 315–357 (2004) · Zbl 1078.53007 · doi:10.4007/annals.2004.160.315
[16] Li, P., Yau, S.T.: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math. 69(2), 269–291 (1982) · Zbl 0503.53042 · doi:10.1007/BF01399507
[17] Montiel, S.: Willmore two-spheres in the four-sphere. Trans. Am. Math. Soc. 352(10), 4469–4486 (2000) · Zbl 0961.53035 · doi:10.1090/S0002-9947-00-02571-X
[18] Müller, S., Sverák, V.: On surfaces of finite total curvature. J. Differ. Geom. 42(2), 229–258 (1995) · Zbl 0853.53003
[19] Pinkall, U., Sterling, I.: Willmore surfaces. Math. Intell. 9(2), 38–43 (1987) · Zbl 0616.53049 · doi:10.1007/BF03025897
[20] Rivière, T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168(1), 1–22 (2006) · Zbl 1128.58010 · doi:10.1007/s00222-006-0023-0
[21] Rusu, R.: An algorithm for the elastic flow of surfaces. Interfaces Free Bound. 7, 229–239 (2005) · Zbl 1210.35149 · doi:10.4171/IFB/122
[22] Simon, L.: Lectures on Geometric Measure Theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3. Australian National University, Centre for Mathematical Analysis, Canberra (1983) · Zbl 0546.49019
[23] Simon, L.: Existence of surfaces minimizing the Willmore functional. Commun. Anal. Geom. 1(2), 281–326 (1993) · Zbl 0848.58012
[24] Simonett, G.: The Willmore flow near spheres. Differ. Integral Equ. 14(8), 1005–1014 (2001) · Zbl 1161.35429
[25] Tartar, L.: Remarks on oscillations and Stokes’ equation. In: Macroscopic Modelling of Turbulent Flows (Nice, 1984). Lect. Notes Phys., vol. 230, pp. 24–31. Springer, Berlin (1985)
[26] Tartar, L.: Imbedding theorems of Sobolev spaces into Lorentz spaces. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 1(3), 479–500 (1998) · Zbl 0929.46028
[27] Thomsen, G.: Über konforme Geometrie, I. Grundlagen der konformen Flächentheorie. Abh. Math. Semin. Univ. Hamburg, pp. 31–56 (1923) · JFM 49.0530.02
[28] Toro, T.: Surfaces with generalized second fundamental form in L2 are Lipschitz manifolds. J. Differ. Geom. 39, 65–101 (1994) · Zbl 0806.53020
[29] Toro, T.: Geometric conditions and existence of bilipschitz parametrisations. Duke Math. J. 77(1), 193–227 (1995) · Zbl 0847.42011 · doi:10.1215/S0012-7094-95-07708-4
[30] Weiner, J.: On a problem of Chen, Willmore, et al. Indiana Univ. Math. J. 27(1), 19–35 (1978) · doi:10.1512/iumj.1978.27.27003
[31] Wente, H.C.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26, 318–344 (1969) · Zbl 0181.11501 · doi:10.1016/0022-247X(69)90156-5
[32] 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
[33] Willmore, T.J.: Note on embedded surfaces. An. Ştiinţ. Univ. Al. I. Cuza Iaşi., Ser. Nouă, Mat. 11B, 493–496 (1965)
[34] Willmore, T.J.: Riemannian Geometry. Oxford University Press, New York (1997) · Zbl 0797.53002
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