Three-manifolds with positive Ricci curvature

TLDR

The authors formulate and analyze a weakly parabolic system of evolution equations for the curvature tensor, employing integrability conditions, interpolation inequalities, and derivative estimates to establish long‑time existence and control of normalized flows in three dimensions. They prove that the flow admits a short‑time solution, preserves positive Ricci curvature while pinching eigenvalues and controlling the scalar curvature gradient, and ultimately converges exponentially to a metric of constant curvature.

Abstract

The evolution equation 259 4. Solution for a short time 260 5. Evolution equations with an integrability condition 262 6. Weakly parabolic linear systems 265 7. Evolution of the curvature 273 8. Curvature in dimension three 276 9. Preserving positive Ricci curvature .27910.Pinching the eigenvalues 283 11.The gradient of the scalar curvature 286 12. Interpolation inequalities for tensors 291 13.Higher derivatives of the curvature .29414.Long time existence 296 15.Controlling R^/R^ 299 16.Estimating the normalized equation 300 17. Exponential convergence 301