We know

- Laplace equation (elliptic equations) $ Δ u = 0$
- Heat equation (parabolic equations) $u_t − Δu = 0$
- Wave equation (hyperbolic equations) $u_{tt} − Δu = 0$

we have - Hyperbolic geometric ﬂow (hyperbolic equations)

$\frac{∂^2}{∂t^2}g_{ij}(t)=-2R_{ij}$

Do any body know the short time existence for heat equation **Hyperbolic Ricci flow**, when our Manifold is non-compact?

and second question: Hyperbolic Ricci flow is invariant under first Chern class?