Abstract

We show that Hamilton’s Ricci flow and the static Einstein vac-uum equations are closely connected by the following system of geometric evolution equations: ∂tg = −2Rc(g) + 2αndu ⊗ du, ∂tu = Δgu, where g(t) is a Riemannian metric, u(t) a scalar function and αn a constant depending only on the dimension n ≥ 3. This provides an interesting and useful link from problems in low-dimensional topology and geometry to physical questions in general relativity. 1.