An elliptic boundary value problem for $G_2$-structures

Donaldson, Simon

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.

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

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