Scalar curvature along Ebin geodesics

TL;DR

This paper proves that in dimensions five and higher, most Ebin geodesics starting from a fixed metric lead to scalar curvature tending to negative infinity as time progresses.

Contribution

It establishes that for generic initial directions, Ebin geodesics exhibit unbounded negative scalar curvature in high dimensions.

Findings

For dimensions ≥ 5, generic geodesics have scalar curvature diverging to -∞.

Open and dense subset of initial directions leads to unbounded negative curvature.

Results hold in the smooth topology for the space of metrics with fixed volume density.

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.