It is often said that the condition of being a positive threeform (in the context of G2 Holonomy Manifolds) is an open condition. My understanding of this statement is that if $\phi$ is a positive threeform and $\phi'$ is a threeform $C_{0}$ close to $\phi$ then $\phi'$ is also a positive threeform. Mathematically this can be expressed via
$\mid \phi-\phi'\mid_{C0} \leq \epsilon$
Where here the norm is given by that of the metric associated to the original g2 threeform. My question is what does $\epsilon$ depend on? I would figure that it does not depend on $\phi'$ but it could depend on $\phi$ or the manifold itself. However, positivity is basically a pointwise condition so I could see how it would be possible for $\epsilon$ to be computed from a computation in $\mathbb{R}^{7}$. Assuming that $\epsilon$ does depend on $\phi$ is it possible to calculate/estimate? I am paricularly interested in the case where the new three-form is of the form $\phi'=\phi+d\beta$ for some two-form $\beta$ in which the argument of the norm is simply d$\beta$
Remember a stable 3-form $\phi$ actually determines a Riemannian metric $g$ since we have the conformal class by $$ i_v\phi\wedge i_w\phi\wedge\phi=g(v,w)\,\mathrm{dvol}_g. $$ So the norm $\lvert\cdot\rvert_{C^0}$ is actually taken with respect to the metric determined by $\phi$. Hence it reduces to pointwise $\mathbb{R}^7$, with $\epsilon>0$ since stable 3-forms has dimension $\dim GL^+(7,\mathbb{R})-\dim G_2=35$, the same dimension as $\bigwedge^3\mathbb{R}^7$. If you want to compute an explicit $\epsilon$, you proceed to bound everything you need in inverse function theorem (i.e., $\lVert f(x)-f(y)\rVert$, $Df$, etc.). However, in practice we just need existence not the numerical value of an $\epsilon$.