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Geometry and Physics: Volume IA Festschrift in honour of Nigel Hitchin$
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Andrew Dancer, Jørgen Ellegaard Andersen, and Oscar García-Prada

Print publication date: 2018

Print ISBN-13: 9780198802013

Published to Oxford Scholarship Online: December 2018

DOI: 10.1093/oso/9780198802013.001.0001

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Boundary Value Problems in Dimensions 7, 4 and 3 Related to Exceptional Holonomy

Boundary Value Problems in Dimensions 7, 4 and 3 Related to Exceptional Holonomy

Chapter:
(p.115) 6 Boundary Value Problems in Dimensions 7, 4 and 3 Related to Exceptional Holonomy
Source:
Geometry and Physics: Volume I
Author(s):

Simon Donaldson

Publisher:
Oxford University Press
DOI:10.1093/oso/9780198802013.003.0006

The variational point of view on exceptional structures in dimensions 6, 7 and 8 is one of Nigel Hitchin’s seminal contributions. One feature of this point of view is that it motivates the study of boundary value problems, for structures with prescribed data on a boundary. This chapter considers the case of 7 dimensions and G2 structures. It briefly reviews a general framework and then goes on to examine in more detail symmetry reductions to dimensions 4 and 3. In the latter case, the chapter presents an interesting variational problem related to the real Monge–Ampère equation and describes a generalization of this.

Keywords:   G2 structure, Monge–Ampère equation, variational problem, boundary value problem

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