Global Conformal Invariants of Submanifolds

Mondino, Andrea; Nguyen, Huy T.

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2663-2695.

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.

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.

Classification : 53A30, 53C40, 53A55
Keywords: Conformal Invariants, Submanifolds, Willmore energy
Mot clés : Invariants conformes, sous-variétés, énergie de Willmore

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