A Note on Decreasing Scalar Curvature From Flat Metrics

Abstract

Abstract. We obtain C ∞ -continuous paths of explicit Riemannian metrics gt, 0 ≤ t < ε, whose scalar curvatures s(gt) decrease, where g0 is a flat metric, i.e. a metric with vanishing curvature. Most of them can exist on tori of dimension ≥ 3. Some of them yield scalar curvature decrease on a ball in the Euclidean space.