Nonembedding and nonextension results in special holonomy

TL;DR

This paper reviews methods for constructing metrics with special holonomy using exterior differential systems, highlighting issues with well-posedness of hypersurface 'flows' and providing counterexamples to existence.

Contribution

It demonstrates that hypersurface 'flows' in special holonomy constructions are often ill-posed and presents counterexamples to the existence of smooth solutions.

Findings

Hypersurface 'flows' are not generally well-posed for smooth initial data.

Counterexamples to the existence of smooth solutions are constructed.

The review connects exterior differential systems with geometric flow interpretations.

Abstract

Constructions of metrics with special holonomy by methods of exterior differential systems are reviewed and the interpretations of these construction as `flows' on hypersurface geometries are considered. It is shown that these hypersurface 'flows' are not generally well-posed for smooth initial data and counterexamples to existence are constructed.