Let $(M^n,g)$ be a closed Riemannian manifold. In this paper, B. Kotschwar proved that the Ricci flow $g(t)$ with initial condition $g(0) = g$ is real analytic with respect to the time variable, for $t \in (0, T)$.
I have two questions:
Is it true that the Ricci flow is real analytic in the time variable at $t = 0$? If not in this generality, is it true when $M^2 \subset \mathbb{R}^3$ is an embedded (convex, maybe) sphere?
Is it true that the normalized Ricci flow is real analytic in the time variable, for $t \in (0, T)$? And for $t = 0$?
Just to recall that the normalized Ricci flow is given by the following equation:
$$ \frac{d}{dt} g(t) = -2 \operatorname{Ric}_{g(t)} + \frac{2}{n} \left( \frac{\int_{M} R_{g(t)} \, dV_t}{\int_{M} 1 \, dV_t} \right) g(t),$$
where $R_{g(t)}$ is the scalar curvature of $(M,g(t))$ and $dV_t$ is the volume element of $(M, g(t))$.