Time-Analyticity of solutions to the ricci flow

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Abstract

We prove that if g(t) is a smooth, complete solution to the Ricci flow of uniformly bounded curvature on M ×[0,Ω], then the correspondence t ↦ g(t) is real-analytic at each t0 ∈ (0,Ω). 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 x0 ∈M and t0 ∈ (0,Ω), there exist local coordinates x = xi on a neighborhood U ⊂ M of x0 in which the representation gij (x,t) of the metric is real-analytic in both x and t on some cylinder U ×(t0 −∈, t0 +∈).

Original languageEnglish (US)
Pages (from-to)535-576
Number of pages42
JournalAmerican Journal of Mathematics
Volume137
Issue number2
DOIs
StatePublished - Jan 1 2015

ASJC Scopus subject areas

  • Mathematics(all)

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