Scalar curvature along Ebin geodesics

Abstract: Let $M$ be a smooth, compact manifold and let $\mathcal{N}{\mu}$ denote the set of Riemannian metrics on $M$ with smooth volume density $\mu$. For a given $g_0\in \mathcal{N}{\mu}$, we show that if $\dim(M)\ge 5$, then there exists an open and dense subset $\mathcal{Y}{g_0} \subset T{g_0} \mathcal{N}{\mu}$ (in the $C^{\infty}$ topology) so that for each $h\in \mathcal{Y}{g_0}$, the $(\mathcal{N}{\mu},L^2)$ Ebin geodesic $\gamma_h(t)$ with $\gamma_h(0)=g_0$ and $\gamma_h'(0)=h$ satisfies $\lim{t \to \infty}$ $R(\gamma_h(t))=-\infty$, uniformly.