Morse functions on the moduli space of G 2 structures Sung

The moduli space of complex structures on a compact Riemann surface of genus 1 or ≥ 2 can be identified with the deformation space of Riemannian metrics of constant curvature 0 or −1 respectively, while the latter definition naturally gives rise to the Weil-Peterson metric. Let M be a compact, oriented, and spin manifold of dimension 7. Then M admits a differential 3-form φ of generic type called a positive( or definite) 3-form(Section 2)[Br]. In dimension 7, such φ determines a unique Riemannian metric gφ and an orientation on M . φ is called a G2 form if dφ = 0, d ∗φ φ = 0, and the orientation determined by φ agrees with the given orientation of M , where ∗φ is the Hodge star operator with respect to gφ. Since the stabilizer of a positive 3-form in the Euclidean space R is isomorphic to compact simple Lie group G2, a G2 form is equivalent to a torsion free G2 structure on M . Throughout this paper, a G2 structure would mean a torsion free G2 structure. It is known that the moduli space of G2 structures, denoted by M, is a smooth manifold of dimension b = dim H(M,R). When M has full holonomy G2, or equivalently b = dim H(M,R)= 0, a connected component of M coincides with the (Ricci flat) Einstein deformation space of the underlying G2 metric (the property for a Ricci flat metric to support a parallel spinor is preserved under Einstein deformation). In this perspective, one of the motivation for the present work is to examine the analogy or the difference between classical Teichmuller theory and the deformation of G2 structures [Tr]. Another motivation comes from the question : Can one find the best G2 form (structure) on a given manifold ? A natural condition would be to require a G2 form φ to satisfy [∗φφ] = −p1(M), where p1(M) ∈ H(M,R) is (nonzero) Pontryagin class.