Morse functions on the moduli space of $G_2$ structures
| dc.creator | Wang, Sung Ho | |
| dc.date | 2002-10-04 | |
| dc.date | 2003-05-04 | |
| dc.date.accessioned | 2026-07-07T04:51:37Z | |
| dc.date.available | 2026-07-07T04:51:37Z | |
| dc.description | Let $ \mathfrak{M}$ be the moduli space of torsion free $ G_2$ structures on a compact 7-manifold $ M$, and let $ \mathfrak{M}_1 \subset \mathfrak{M}$ be the $ G_2$ structures with volume($M$) $=1$. The cohomology map $ π^3: \mathfrak{M} \to H^3(M, R)$ is known to be a local diffeomorphism. It is proved that every nonzero element of $ H^4(M, R) = H^3(M, R)^*$ is a Morse function on $ \mathfrak{M}_1 $ when composed with $ π^3$. When dim $H^3(M, R) = 2$, the result in particular implies $ π^3$ is one to one on each connected component of $ \mathfrak{M}$. Considering the first Pontryagin class $ p_1(M) \in H^4(M, R)$, we formulate a compactness conjecture on the set of $ G_2$ structures of volume($M$) $=1$ with bounded $L^2$ norm of curvature, which would imply that every connected component of $ \mathfrak{M}$ is contractible. We also observe the locus $ π^3(\mathfrak{M}_1) \subset H^3(M, R)$ is a hyperbolic affine sphere if the volume of the torus $ H^3(M, R) / H^3(M, Z)$ is constant on $ \mathfrak{M}_1$. | |
| dc.description | 17 pages | |
| dc.identifier | https://arxiv.org/abs/math/0210054 | |
| dc.identifier | http://arxiv.org/abs/math/0210054 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/65168 | |
| dc.subject | Differential Geometry | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 53C25 | |
| dc.title | Morse functions on the moduli space of $G_2$ structures | |
| dc.type | text |