An elliptic boundary value problem for $G_{2}$ structures

TL;DR

This paper demonstrates that the $G_{2}$ holonomy equation with boundary conditions is elliptic, enabling a deformation theory that includes constructing $G_{2}$ cobordisms between small Calabi-Yau 3-fold deformations.

Contribution

It establishes the ellipticity of the $G_{2}$ boundary value problem and develops a deformation theory including the existence of $G_{2}$ cobordisms.

Findings

Proves ellipticity of the $G_{2}$ boundary value problem.

Develops a deformation theory for $G_{2}$ structures.

Shows existence of $G_{2}$ cobordisms between small Calabi-Yau deformations.

Abstract

We show that the $G_{2}$ holonomy equation on a manifold with boundary, with prescribed 3-form on the boundary, is elliptic. The main point is to set up a suitable linear elliptic boundary value problem. This result leads to a deformation theory. In particular we establish the existence of certain $G_{2}$ cobordisms between two small deformations of a Calabi-Yau 3-fold.