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Time-analyticity of solutions to the Ricci flow
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 137, Number 2, April 2015
- pp. 535-576
- 10.1353/ajm.2015.0012
- Article
- Additional Information
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We prove that if $g(t)$ is a smooth, complete solution to the Ricci flow of uniformly bounded curvature on
$M\times[0, \Omega]$, then the correspondence $t\mapsto g(t)$ is real-analytic at each $t_0\in (0, \Omega)$.
The analyticity is a consequence of classical Bernstein-type estimates on the temporal and spatial derivatives
of the curvature tensor, which we further use to show that, under the above global hypotheses, for any $x_0\in M$
and $t_0\in (0,\Omega)$, there exist local coordinates $x = x^i$ on a neighborhood $U\subset M$ of $x_0$ in which
the representation $g_{ij}(x, t)$ of the metric is real-analytic in both $x$ and $t$ on some cylinder
$U\times (t_0 -\epsilon, t_0 + \epsilon)$.