Global Conformal Invariants of Submanifolds
Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2663-2695.

The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In codimension one we classify such invariants, showing that under a structural hypothesis (more precisely we assume the integrand depends separately on the intrinsic and extrinsic curvatures, and not on their derivatives) the integrand can only consist of an intrinsic scalar conformal invariant, an extrinsic scalar conformal invariant and the Chern–Gauss–Bonnet integrand. In particular, for codimension one surfaces, we show that the Willmore energy is the unique global conformal invariant, up to the addition of a topological term (the Gauss curvature, giving the Euler Characteristic by the Gauss Bonnet Theorem). A similar statement holds also for codimension two surfaces, once taking into account an additional topological term given by the Chern–Gauss–Bonnet integrand of the normal bundle. We also discuss existence and properties of natural higher dimensional (and codimensional) generalizations of the Willmore energy.

Le but de cet article est d’étudier la structure algébrique des invariants conformes globaux des sous-variétés. Ceux-ci sont définis comme étant des intégrales conformément invariantes des scalaires géométriques du fibré tangent et normal. Un exemple célèbre d’un invariant conforme global est l’énergie de Willmore d’une surface. En codimension un, nous classons ces invariants, montrant que sous une hypothèse structurelle (plus précisément nous supposons que l’intégrande dépend séparément des courbures intrinsèque et extrinsèque , et non de leurs dérivées) l’intégrande ne peut être constitué que d’un invariant conforme scalaire intrinsèque, d’un invariant conforme scalaire extrinsèque ou de l’intégrande de Chern–Gauss–Bonnet. En particulier, pour les surfaces de codimenson 1, nous montrons que l’énergie de Willmore est l’unique invariant conforme global, jusqu’à l’addition d’un terme topologique (la courbure de Gauss, donnant la caractéristique d’Euler par le théorème de Gauss Bonnet). Un résultat similaire est également valable pour les surfaces de codimension deux, en prenant en compte un terme topologique supplémentaire donné par l’intégrande de Chern–Gauss–Bonnet du faisceau normal. Nous discutons également de l’existence et des propriétés des généralisations naturelles en dimensions (et codimensions) supérieures de l’énergie de Willmore.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3220
Classification: 53A30,  53C40,  53A55
Keywords: Conformal Invariants, Submanifolds, Willmore energy
Mondino, Andrea 1; Nguyen, Huy T. 2

1 University of Warwick Mathematics Institute CV4 7AL Coventry (UK)
2 Queen Mary University of London School of Mathematical Sciences Mile End Road E1 4NS London (UK)
@article{AIF_2018__68_6_2663_0,
     author = {Mondino, Andrea and Nguyen, Huy T.},
     title = {Global {Conformal} {Invariants} of {Submanifolds}},
     journal = {Annales de l'Institut Fourier},
     pages = {2663--2695},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     doi = {10.5802/aif.3220},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3220/}
}
TY  - JOUR
TI  - Global Conformal Invariants of Submanifolds
JO  - Annales de l'Institut Fourier
PY  - 2018
DA  - 2018///
SP  - 2663
EP  - 2695
VL  - 68
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3220/
UR  - https://doi.org/10.5802/aif.3220
DO  - 10.5802/aif.3220
LA  - en
ID  - AIF_2018__68_6_2663_0
ER  - 
%0 Journal Article
%T Global Conformal Invariants of Submanifolds
%J Annales de l'Institut Fourier
%D 2018
%P 2663-2695
%V 68
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3220
%R 10.5802/aif.3220
%G en
%F AIF_2018__68_6_2663_0
Mondino, Andrea; Nguyen, Huy T. Global Conformal Invariants of Submanifolds. Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2663-2695. doi : 10.5802/aif.3220. https://aif.centre-mersenne.org/articles/10.5802/aif.3220/

[1] Alexakis, Spyros On the decomposition of global conformal invariants, II, Adv. Math., Volume 206 (2006) no. 2, pp. 466-502 | Zbl: 1124.53013

[2] Alexakis, Spyros On the decomposition of global conformal invariants, I, Ann. Math., Volume 170 (2009) no. 3, pp. 1241-1306 | Zbl: 1190.53028

[3] Alexakis, Spyros The decomposition of global conformal invariants, IV: A proposition on local Riemannian invariants, Adv. Math., Volume 225 (2011) no. 2, pp. 515-597 | Zbl: 1198.53034

[4] Alexakis, Spyros The decomposition of Global Conformal Invariants: Some technical proofs. I, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 7 (2011), 019, 41 pages (Art. ID 019, 41 p.) | Zbl: 1218.53038

[5] Alexakis, Spyros The decomposition of Global Conformal Invariants, Annals of Mathematics Studies, 182, Princeton University Press, 2012, v+449 pages | Zbl: 1258.53003

[6] Alexakis, Spyros The decomposition of Global Conformal Invariants: Some technical proofs 2, Pac. J. Math., Volume 260 (2012) no. 1, pp. 1-88 | Zbl: 1271.53015

[7] Bailey, Toby N.; Eastwood, Michael G.; Graham, C. Robin Invariant theory for conformal and CR geometry, Ann. Math., Volume 139 (1994) no. 3, pp. 491-552 | Zbl: 0814.53017

[8] Bernard, Yann; Rivière, Tristan Energy Quantization for Willmore Surfaces and Applications, Ann. Math., Volume 180 (2014) no. 1, pp. 87-136 | Zbl: 1325.53014

[9] Blaschke, Wilhelm Vorlesungen über Differentialgeometrie, 3, Grundlehren der Mathematischen Wissenschaften, 1, Springer, 1929 | Zbl: 56.0588.07

[10] Carlotto, Alessandro; Mondino, Andrea Existence of Generalized Totally Umbilic 2-Spheres in Perturbed 3-Spheres, Int. Math. Res Not., Volume 2014 (2014) no. 21, pp. 6020-6052 | Zbl: 1303.53079

[11] Deser, S.; Schwimmer, Adam Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B, Volume 309 (1993), pp. 279-284

[12] Eastwood, Michael G. Notes on conformal geometry, The Proceedings of the 19th Winter School “Geometry and Physics” (Supplemento ai Rendiconti del Circolo Matemàtico di Palermo), Volume 43, Circolo Matemàtico di Palermo, 1996, pp. 57-76 | Zbl: 0911.53020

[13] Gilkey, Peter B. Local Invariants of an Embedded Riemannian Manifold, Ann. Math., Volume 102 (1975) no. 2, pp. 198-203

[14] Kuwert, Ernst; Mondino, Andrea; Schygulla, Johannes Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds, Math. Ann., Volume 359 (2014) no. 1-2, pp. 379-425 | Zbl: 1295.53028

[15] Kuwert, Ernst; Schätzle, Reiner Removability of isolated singularities of Willmore surfaces, Ann. Math., Volume 160 (2004) no. 1, pp. 315-357 | Zbl: 1078.53007

[16] Lamm, Tobias; Metzger, Jan; Schulze, Felix Foliations of asymptotically flat manifolds by surfaces of Willmore type, Math. Ann., Volume 350 (2011), pp. 1-78 | Zbl: 1222.53028

[17] Li, Peter; Yau, Shing-Tung A New Conformal Invariant and its Applications to the Willmore Conjecture and the First Eigenvalue on Compact Surfaces, Invent. Math., Volume 69 (1982), pp. 269-291 | Zbl: 0503.53042

[18] Marques, Fernando C.; Neves, André Min-Max theory and the Willmore conjecture, Ann. Math., Volume 179 (2014) no. 2, pp. 683-782 | Zbl: 1297.49079

[19] Mondino, Andrea The conformal Willmore Functional: a perturbative approach, Journal of Geometric Analysis, Volume 23 (2013) no. 2, pp. 764-811

[20] Mondino, Andrea; Rivière, Tristan Willmore spheres in compact Riemannian manifolds, Adv. Math., Volume 232 (2013) no. 1, pp. 608-676 | Zbl: 1294.30046

[21] Montiel, Sebastiań; Urbano, Francisco A Willmore functional for compact surfaces in the complex projective plane, J. Reine Angew. Math., Volume 546 (2002), pp. 139-154 | Zbl: 0999.53040

[22] Rivière, Tristan Analysis aspects of Willmore surfaces, Invent. Math., Volume 174 (2008) no. 1, pp. 1-45 | Zbl: 1155.53031

[23] Simon, Leon Existence of surfaces minimizing the Willmore functional, Commun. Anal. Geom., Volume 1 (1993), pp. 281-326 | Zbl: 0848.58012

[24] Willmore, Tom J. Riemannian Geometry, Oxford Science Publications, Clarendon Press, 1993, xi+318 pages | Zbl: 0797.53002

Cited by Sources: