Topping's existence conjecture for Ricci flow on complete noncompact 3-manifolds

Let (M3,g0)(M^3,g_0) be a complete noncompact 33-dimensional Riemannian manifold with nonnegative, possibly unbounded, Ricci curvature

Ric(g0)0.\operatorname{Ric}(g_0)\geq 0.

Topping's Ricci-flow existence conjecture. The Ricci flow equation

gt=2Ric(g),g(0)=g0,\frac{\partial g}{\partial t}=-2\operatorname{Ric}(g),\qquad g(0)=g_0,

has a corresponding smooth solution g(t)g(t) on M3×[0,T)M^3\times[0,T) for some T>0T>0, and g(t)g(t) is complete and has nonnegative Ricci curvature for each t[0,T)t\in[0,T). This extends Shi's existence theory beyond the bounded-curvature setting and is a special case of a conjecture attributed to Topping; the statement concerns preservation of completeness and nonnegative Ricci curvature despite initially unbounded curvature.

Sources & referencesView supporting material

Primary source

Albert Chau and Adam Martens, “Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows”, arXiv:2307.08088 (2024).

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