TY - JOUR

T1 - Time-Analyticity of solutions to the ricci flow

AU - Kotschwar, Brett

N1 - Publisher Copyright:
© 2015 Johns Hopkins University Press. All rights reserved.

PY - 2015

Y1 - 2015

N2 - 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 +∈).

AB - 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 +∈).

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U2 - 10.1353/ajm.2015.0012

DO - 10.1353/ajm.2015.0012

M3 - Article

AN - SCOPUS:84927658285

VL - 137

SP - 535

EP - 576

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 2

ER -