The moduli space of torsion-free G2 structures

Let M be the moduli space of torsion-free G2 structures on a compact oriented G2 manifold M. The natural cohomology map π3:M→H3(M,ℝ) is known to be a local diffeomorphism [Compact Manifolds with Special Holonomy, Oxford University Press, 2000]. Let M1⊂M be the subset of G2 structures with volume (M)=1. We show every nonzero element of H4(M,ℝ)=H3(M,ℝ)* is a Morse function on M1 when composed with π3, and we compute its Hessian. The result implies a special case of Torelli's theorem: if H1(M,ℝ)=0 and dimH3(M,ℝ)=2, the cohomology map π3:M→H3(M,ℝ) is one to one on each connected component of M. We formulate a compactness conjecture on the set of G2 structures of volume (M)=1 with bounded L2 norm of curvature. If this conjecture were true, it would imply that every connected component of M is contractible, and that every compact G2 manifold supports a G2 structure whose fundamental 4-form represents the negative of the (nonzero) first Pontryagin class of M. We also observe that when H1(M,ℝ)=0, and the volume of the torus H3(M,ℝ)/H3(M,ℤ) is constant along M1, the locus π3(M1)⊂H3(M,ℝ) is a hyperbolic affine sphere.