Time-analyticity of solutions to the Ricci flow

2012/10/10 by Brett Kotschwar, Kotschwar, Brett
Mathematics · Physics and Astronomy · #53C44 #58J35 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1210.3083

openalex publication_date 2012/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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×[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 g_ij(x, t) of the metric is real-analytic in both x and t on some cylinder U× (t₀ -ε, t₀ + ε).

Citations

• Analyticity at the boundary of solutions of nonlinear second‐order parabolic equations
• Analyticity up to the boundary of solutions of nonlinear parabolic equations
• An energy approach to the problem of uniqueness for the Ricci flow

Related

• A local curvature estimate for the Ricci flow
• On the regularity of Ricci flows coming out of metric spaces
• A note on Kaehler-Ricci flow
• Curvature Estimates for the Ricci Flow II
• The Long-Time Behavior of the Ricci Tensor Under The Ricci Flow