Stability of Euclidean space under Ricci flow
By Oliver C. Schnurer, F. Schulze, Miles Simon. 2007
73 citations recorded by the source corpus.
We study the Ricci flow for initial metrics which are C^0 small perturbations of the Euclidean metric on R^n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In proving this stability result, we introduce a monotone integral quantity which measures the deviation of the evolving metric from the Euclidean metric. We also investigate the convergence of the diffeomorphisms relating Ricci harmonic map heat flow to Ricci flow.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/4ae4ed62c113d54cbea07aa61491a8f8b2d5b690
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In Applied Mathematics
- T. Lamm
- Herbert Koch
- Changyou Wang
- D. Tataru
- Carlos E. Kenig
- Pierre Germain
- Jean-Yves Chemin
- Isabelle Gallagher
In Astronomy and Astrophysics
In Geometry and Topology
Citations in the corpus
- Geometric flows with rough initial data (2009, background): “[20] M. Simon.”
- Ricci flow coupled with harmonic map flow (2009, unknown): “For the dual Ricci-Harmonic flow, the evolution equations in coordinate form, using only the fixed connection ∇̂ of the background metric h, are (see e.g. Simon [39]) ∂ ∂t g̃ij = g̃ kℓ∇̂k∇̂ℓg̃ij − g̃kℓg̃iphpqR̂jkqℓ − g̃kℓg̃jphpqR̂ikqℓ + 12 g̃ kℓg̃pq ( ∇̂ig̃pk∇̂j g̃qℓ + 2∇̂kg̃ip∇̂q g̃jℓ − 2∇̂kg̃ip∇̂ℓ”