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Global conformal invariants of submanifolds
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Mondino, Andrea and The Nguyen, Huy (2018) Global conformal invariants of submanifolds. Annales de l'Institute Fourier, 68 (6). pp. 2663-2695. doi:10.5802/aif.3220 ISSN 1777-5310.
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Official URL: https://doi.org/10.5802/aif.3220
Abstract
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
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Library of Congress Subject Headings (LCSH): | Submanifolds, Scalar field theory, Riemannian manifolds | ||||||
Journal or Publication Title: | Annales de l'Institute Fourier | ||||||
Publisher: | Annales de l’Institut Fourier | ||||||
ISSN: | 1777-5310 | ||||||
Official Date: | 22 November 2018 | ||||||
Dates: |
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Volume: | 68 | ||||||
Number: | 6 | ||||||
Page Range: | pp. 2663-2695 | ||||||
DOI: | 10.5802/aif.3220 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Date of first compliant deposit: | 7 February 2018 | ||||||
Date of first compliant Open Access: | 7 February 2018 | ||||||
RIOXX Funder/Project Grant: |
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