Time-analyticity of solutions to the Ricci flow

TL;DR

This paper proves that solutions to the Ricci flow with bounded curvature are real-analytic in both space and time around any positive time, using Bernstein-type estimates to establish the analyticity of the metric tensor.

Contribution

It establishes the real-analyticity of Ricci flow solutions in space and time under bounded curvature conditions, extending previous regularity results.

Findings

Solutions are real-analytic in space and time near any positive time.

Bernstein-type estimates are used to prove analyticity.

Local coordinates exist where the metric is real-analytic in both variables.

Abstract

In this paper, 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 $U\times (t_0 -\epsilon, t_0 + \epsilon)$.