Morse functions on the moduli space of G 2 structures

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, while the latter definition natually gives rise to the Weil-Peterson metric. Let M be a compact manifold of domension 7 with an integrable G2 structure, i.e., a differential 3-form φ that satisfies dφ = 0, and d ∗φ φ = 0. It is known that the moduli space of G2 structures, say M, is a smooth manifold of dimension b 3 = dimH(M,R). Moreover, when M has full holonomy G2, or equivalently b 1 = 0, each connected component of M coincides with the (Ricci flat) Einstein defomation space of the G2 metrics (the property for a Ricci flat metric to support a parallel spinor is preserved under Einstein deformation). In this regard, one of the motivation for the present work is to examine the analogy or the difference betweeen classical Teichmuller theory and the theory of deformation of G2 structures [Tr]. Another motivation comes from the question : Can one find the best G2 form (structure) on a given manifold ? An obvious candidate would be a G2 form φ such that [∗φφ] = −p1(M), where p1(M) ∈ H(M,R) is the Pontryagin class. The purpose of this paper is to show that any nonzero linear(height) function β ∈ H(M,R) = H3(M,R)∗ composed with the porjection map M → H(M,R) is a Morse function on M1, where M1 ⊂ M is the moduli space of G2 structures of volume 1. In particular, when b = 0 and β = −p1(M), every critical points, if exists, is a positive local minimum. As a corollary, p1(M) 6= 0 implies each connected component of M1 is noncompact. We also prove Torelli theorem in case b = 0 and b = 2 : the cohomology map M → H(M,R) is one to one on each connected component.