Morse functions on the moduli space of G₂ structures

2002/10/04 by Sung Ho Wang, Wang, Sung Ho
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.AP #math.DG #msc:53C25

paper · pdf · doi:10.48550/arxiv.math/0210054

17 pages

arxiv created 2003/05/04 · arxiv updated 2009/11/30

Abstract

Let \mathfrakM be the moduli space of torsion free G₂ structures on a compact 7-manifold M, and let \mathfrakM₁ ⊂ \mathfrakM be the G₂ structures with volume(M) =1. The cohomology map π³: \mathfrakM → H³(M, R) is known to be a local diffeomorphism. It is proved that every nonzero element of H⁴(M, R) = H³(M, R)^* is a Morse function on \mathfrakM₁ when composed with π³. When dim H³(M, R) = 2, the result in particular implies π³ is one to one on each connected component of \mathfrakM. Considering the first Pontryagin class p₁(M) ∈ H⁴(M, R), we formulate a compactness conjecture on the set of G₂ structures of volume(M) =1 with bounded L² norm of curvature, which would imply that every connected component of \mathfrakM is contractible. We also observe the locus π³(\mathfrakM₁) ⊂ H³(M, R) is a hyperbolic affine sphere if the volume of the torus H³(M, R) / H³(M, Z) is constant on \mathfrakM₁.

Citations

• Einstein metrics and complex structures

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