An elliptic boundary value problem for G 2 -structures
[Un problème de type elliptique avec données au bord pour des structures G 2 ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2783-2809.

Nous montrons que l’équation d’holonomie G 2 sur une variété de dimension 7 à bord, avec 3-forme prescrite sur le bord et modulo l’action de difféomorphismes, est elliptique. Le point clé est de mettre en place une théorie linéaire elliptique adaptée. Avec ce résultat une théorie de la déformation est définie, gouvernée par un espace d’obstruction de dimension finie. Nous discutons les conditions pour lesquelles cet espace d’obstruction est trivial, et donnons une application en démontrant l’existence de certains G 2 -cobordismes entre deux petites déformations d’une variété Calabi–Yau de dimension 3.

We show that the G 2 holonomy equation on a seven-dimensional manifold with boundary, with prescribed 3-form on the boundary and modulo the action of diffeomorphisms, is elliptic. The main point is to set up a suitable linear elliptic theory. This result leads to a deformation theory, governed by a finite-dimensional obstruction space. We discuss conditions under which this obstruction space vanishes and as one application we establish the existence of certain G 2 cobordisms between two small deformations of a Calabi–Yau 3-fold.

Publié le :
DOI : 10.5802/aif.3225
Classification : 53C26
Keywords: exceptional holonomy, $G_{2}$-structures, elliptic boundary value problems
Mot clés : Groupe d’holonomie exceptionnelle, structures $G_{2}$, problèmes de type elliptique avec données au bord
Donaldson, Simon 1

1 The Simons Center for Geometry and Physics Stony Brook and Imperial College London (UK)
@article{AIF_2018__68_7_2783_0,
     author = {Donaldson, Simon},
     title = {An elliptic boundary value problem for $G_{2}$-structures},
     journal = {Annales de l'Institut Fourier},
     pages = {2783--2809},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {7},
     year = {2018},
     doi = {10.5802/aif.3225},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3225/}
}
TY  - JOUR
AU  - Donaldson, Simon
TI  - An elliptic boundary value problem for $G_{2}$-structures
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 2783
EP  - 2809
VL  - 68
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.3225/
DO  - 10.5802/aif.3225
LA  - en
ID  - AIF_2018__68_7_2783_0
ER  - 
%0 Journal Article
%A Donaldson, Simon
%T An elliptic boundary value problem for $G_{2}$-structures
%J Annales de l'Institut Fourier
%D 2018
%P 2783-2809
%V 68
%N 7
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.3225/
%R 10.5802/aif.3225
%G en
%F AIF_2018__68_7_2783_0
Donaldson, Simon. An elliptic boundary value problem for $G_{2}$-structures. Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2783-2809. doi : 10.5802/aif.3225. http://archive.numdam.org/articles/10.5802/aif.3225/

[1] Bryant, Robert L. Metrics with exceptional holonomy, Ann. Math. (1987), pp. 525-576 | Zbl

[2] Bryant, Robert L. Some remarks on G 2 -structures, Gökova Geometry and Topology (Gökova, 2004-2005, International Press, 2005, pp. 71-109 | Zbl

[3] Donaldson, Simon K. Floer homology groups in Yang-Mills theory, Cambridge Tracts in Mathematics, 147, Cambridge University Press, 2002, vii+236 pages | Zbl

[4] Donaldson, Simon K. Adiabatic limits of co-associative Kovalev-Lefschetz fibrations, Algebra, geometry, and physics in the 21st century (Progress in Mathematics), Volume 324, Birkhäuser, 2017, pp. 1-29 | Zbl

[5] Donaldson, Simon K. Boundary value problems in dimensions seven, four and three related to exceptional holonomy, Geometry and physics. Vol. I, Oxford University Press, 2018 | DOI

[6] Fine, Joel; Lotay, Jason D.; Singer, Michael The space of hyperkähler metrics on a 4-manifold with boundary, Forum Math. Sigma, Volume 5 (2017), e6, 50 pages (Art. ID e6, 50 p.) | Zbl

[7] Hitchin, Nigel The geometry of three-forms in six dimensions, J. Differ. Geom., Volume 55 (2000) no. 3, pp. 547-576 | Zbl

[8] Hitchin, Nigel Stable forms and special metrics, Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000) (Contemporary Mathematics), Volume 288, American Mathematical Society, 2000, pp. 70-89 | Zbl

[9] Joyce, Dominic Compact Riemannian manifolds with exceptional holonomy G 2 . I, Essays on Einstein manifolds (Surveys in Differential Geometry), Volume 6, International Press, 1996, pp. 291-328 | Zbl

[10] Joyce, Dominic Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, 2000, xii+436 pages | Zbl

[11] Schwarz, Günter Hodge decomposition. A method for solving boundary value problems, Lecture Notes in Mathematics, 1607, Springer, 1995, 155 pages | Zbl

[12] Taylor, Michael E. Pseudodifferential operators, Princeton Mathematical Series, 34, Princeton University Press, 1981, xi+452 pages | Zbl

Cité par Sources :