Ricci flow on a compact surface of constant negative curvature

Question

Let us consider a compact surface of constant negative curvature $-1$ and apply the Ricci flow on it. Will the resulting surfaces for short time also have constant negative curvature? If yes, will the curvature still be equal to $-1$ for short time? Thanks!

Answer 1

For any Einstein manifold (whose satisfies

$$R_{ij} = C g_{ij}$$

for some $C$), the Ricci flow equation becomes

$$\partial_t g_{ij} = -2 R_{ij} = -2C g_{ij}\Rightarrow (g_t)_{ij} = e^{-2Ct} g_{ij}$$

Thus $g_{ij}$ will be scaled and the curvature is changed (to a different constant) too.