10 Singularity models
Blow-up limits at a singularity are the objects this chapter describes, and before any of them can be recognised one has to know they have non-negative curvature. In dimension three that is not automatic — it is the Hamilton–Ivey estimate, and it was missing from this chart entirely.
Let \((M^3, g(t))\) be a Ricci flow on a closed three-manifold, normalised so that the smallest eigenvalue \(\nu \) of the curvature operator satisfies \(\nu \ge -1\) at \(t = 0\). Then at every point and every \(t \ge 0\) at which \(\nu {\lt} 0\),
(Hamilton 1999 §24; Ivey 1993; Morgan–Tian 4.4; Chow–Knopf 6.44.)
Negative curvature is dominated by the scalar curvature, at a rate that degenerates only logarithmically. Rescaling a singularity multiplies \(\operatorname {scal}\) by a factor tending to infinity while the estimate is scale-invariant up to the logarithm, so every blow-up limit has \(\nu \ge 0\): the limits are non-negatively curved. That is exactly the standing hypothesis in Definition 122, and without this theorem no blow-up limit is ever known to satisfy it.
Structurally this is Lemma 89 again — a closed convex set of curvature operators preserved by the ODE \(\dot{\operatorname {Rm}} = \operatorname {Rm}^2 + \operatorname {Rm}^{\# }\), transported to the flow by the tensor maximum principle. Hamilton’s set is
and the whole content at the ODE level is that \(K\) is preserved. (Cao–Zhu, Theorem 2.4.1.) The ODE half is done — Lemma 120 — and so is everything Theorem 78 asks of \(K\): it is closed, convex, and invariant under the ODE (Lemma 119). An earlier version of this node said the transport still needed \(f^{-1}\) on \([-e^2,\infty )\) and its concavity; it does not, and never did. What remains is the flow itself, \(\partial _t \operatorname {Rm}= \Delta \operatorname {Rm}+ Q\) (Lemma 66). For Hamilton’s later improvement with the \(\log (1+t)\) term one additionally needs a form of Theorem 78 for a time-dependent family \(\{ K_t\} \); that is a separate, still-open item.
Along Hamilton’s curvature ODE,
so every lower bound \(\lambda +\mu +\nu \ge c\) is preserved. With \(c = -3\) this is the first of the two inequalities cutting out \(K\).
The second inequality cutting out \(K\) reads \(\lambda +\mu \ge (-\nu )[\log (-\nu )-2]\) whenever \(\nu \le -e^2\), and need only be checked on the boundary. There the defining relation eliminates the logarithm and what is left is polynomial. Writing \(N = -\nu {\gt} 0\) and \(L = \log N\):
if \(\mu \ge 0\), the boundary relation is \(\lambda + \mu = N(L-2)\) and the required inequality \(\dot\lambda + \dot\mu \ge (L-1)\, \dot{\overparen {(-\nu )}}\) becomes, after multiplying by \(N\),
\[ N(\lambda ^2+\mu ^2) + N^3 + \lambda \mu (\lambda +\mu +N) \ge 0 ; \]if \(\mu {\lt} 0\), put \(P = -\mu \), so the ordering \(\nu \le \mu \) is \(P \le N\) and the boundary relation is \(\lambda = P + N(L-2)\); the required inequality becomes
\[ (\lambda ^2 - \lambda P + P^2)(N - P) + P^3 + N^3 \ge 0 . \]
Both hold because every factor is nonnegative — in the second, \(\lambda ^2-\lambda P+P^2 = (\lambda - P/2)^2 + \tfrac 34 P^2\). (Cao–Zhu, Theorem 2.4.1, Cases (i) and (ii); in Lean hamiltonIvey_boundary_of_nonneg and hamiltonIvey_boundary_of_neg.)
Because \(f^{-1}\) takes values in \([e^2,\infty )\), the second condition cutting out \(K\) holds automatically when \(-\nu \le e^2\) and is \(f(-\nu ) \le \lambda +\mu +\nu \) otherwise. So \(K\) can be written with no inverse function at all:
In that form both ends of Hamilton’s argument are elementary.
Entry. An ordered triple with \(\nu \ge -1\) lies in \(K\): the ordering gives \(\lambda +\mu +\nu \ge 3\nu \ge -3\), and \(-\nu \le 1 \le e^2\).
Exit. An ordered triple in \(K\) with \(\nu {\lt} 0\) satisfies \(\operatorname {scal}\ge (-\nu )(\log (-\nu ) - 3)\). Write \(N = -\nu \). For \(N {\gt} e^2\) this is the second condition verbatim. For \(N \le 1\) we have \(\log N \le 0\), so \(f(N) \le -3N\), and the ordering gives \(\operatorname {scal}\ge 3\nu = -3N\). For \(1 \le N \le e^2\) the function \(f\) is antitone — there \(f' = \log N - 2 \le 0\) — so \(f(N) \le f(1) = -3\), and the first condition of \(K\) finishes it. That middle range is the whole reason \(K\) carries the otherwise unmotivated bound \(\lambda +\mu +\nu \ge -3\).
\(K\) is a closed convex subset of \(\mathbb {R}^3\) — which is what Theorem 78 requires of it, and the one thing missing before Lemma 120 can be carried from the ODE to the flow.
Proved (IveyConvex.lean), and without building \(f^{-1}\). The literature restores convexity by rewriting the second condition as \(-\nu \le f^{-1}(\lambda +\mu +\nu )\) and invoking concavity of \(f^{-1}\); an earlier version of these notes recorded that inverse, and its concavity, as the remaining obstacle. It is not needed. The disjunction of Lemma 118 collapses to a single inequality against
because \(f\) is increasing on \([e^2,\infty )\): when \(-\nu \le e^2\) we get \(G(\nu ) = f(e^2) = -e^2 \le -3\), which the first condition of \(K\) already supplies, and otherwise \(G(\nu ) = f(-\nu )\) on the nose (isIveyPinched_iff_iveyG).
\(f\) is convex on \([e^2,\infty )\) because \(f' = \log x - 2\) is monotone there, and increasing there because \(f' \ge 0\); \(\nu \mapsto \max (-\nu , e^2)\) is convex, being a pointwise maximum of two affine functions, and its image is exactly \([e^2,\infty )\). A convex function composed with a convex function that lands where the outer one is monotone is convex, so \(G\) is convex on all of \(\mathbb {R}\) (convexOn_iveyG). The set is then cut out by three half-spaces — \(\nu \le \mu \), \(\mu \le \lambda \), \(\lambda +\mu +\nu \ge -3\) — and the single convex inequality \(G(\nu ) \le \lambda +\mu +\nu \), hence is convex; and closed, \(G\) being continuous.
Invariance, in the same language. An ordered solution of the curvature ODE normalised by \(\nu (0) \ge -1\) stays in \(K\) for all time (IsCurvatureODE.isIveyPinched, and IsCurvatureODE.mem_iveyPinchedSet as membership of the subset of \(\mathbb {R}^3\)): \(R\) is nondecreasing and starts at \(\ge 3\nu (0) \ge -3\), and where \(\nu {\lt} 0\) the second condition is Lemma 120 verbatim, while where \(\nu \ge 0\) the branch \(-\nu \le e^2\) is free. So both of the inputs Theorem 78 needs about \(K\) — closed convex, and invariant under the ODE — are now available; what is still missing for the transport is the flow itself, \(\partial _t \operatorname {Rm}= \Delta \operatorname {Rm}+ Q\) (Lemma 66).
Let \(\lambda \ge \mu \ge \nu \) solve Hamilton’s curvature ODE on \([0,T]\) with \(\nu (0) \ge -1\). Then at every time at which \(\nu {\lt} 0\),
Cao–Zhu check invariance of \(K\) on its boundary, in two cases according to the sign of \(\mu \). The two cases are the same inequality: the hypothesis of the second, \(\lambda = -\mu + N(L-2)\) with \(N = -\nu \), is the hypothesis of the first, \(\lambda +\mu = N(L-2)\), and eliminating \(L\) from either leaves the same polynomial
which is nonnegative under the ordering alone — no boundary relation, no logarithm. So the boundary argument becomes a linear Grönwall comparison: for the defect \(\Psi = \operatorname {scal}- f(-\nu )\) one has the identity
valid wherever \(\nu {\lt} 0\), so \(\Psi ' \ge a\Psi \) with \(a = -(N^2+\lambda \mu )/N\) continuous, and \(\Psi \ge 0\) is preserved.
That \(\nu {\lt} 0\) on the whole interval is not an extra hypothesis: non-negative curvature is preserved by the ODE (\(\dot\nu \ge \lambda \nu \) under the ordering), so a \(\nu \) that is negative at \(t\) was negative on all of \([0,t]\), and where \(\nu \ge 0\) the estimate is vacuous.
A sequence of pointed Ricci flows with uniformly bounded curvature and a uniform injectivity-radius lower bound at the basepoints subconverges, in \(C^\infty \) on compact sets, to a limiting pointed flow. Blow-up limits at a singularity are obtained this way. (Morgan–Tian Chapter 5, five sections: convergence of manifolds, of flows, Gromov–Hausdorff, blow-up limits, splitting limits at infinity.)
Blocked, separately from the PDE gap. Mathlib has the Gromov–Hausdorff distance between compact metric spaces; the pointed, smooth, curvature-bounded convergence theory for manifolds is a different and much larger object, and none of it exists. It is the second independent analytic prerequisite on this road and the first not shared with Hamilton’s theorem.
An ancient solution — defined on \((-\infty , 0]\) — that is \(\kappa \)-noncollapsed, has bounded non-negative curvature operator, and is not flat. These are exactly the models that arise from rescaling a singularity.
In dimension three every \(\kappa \)-solution is, at every point and scale, close to one of a short explicit list: the round shrinking \(S^3/\Gamma \), the round shrinking neck \(S^2 \times \mathbb {R}\), or a capped variant. (Morgan–Tian Chapter 9, eight sections — the asymptotic gradient shrinking soliton, splitting at infinity, classification of gradient shrinking solitons in dimensions 2 and 3, a universal \(\kappa \), asymptotic volume, compactness of the space of \(3\)-dimensional \(\kappa \)-solutions.)
The technical core of Perelman’s second preprint and the longest single chapter of the book.
Curvature control at a point propagates to control on a definite neighbourhood. (Morgan–Tian Chapter 10.)
A separate chapter, proved by contradiction through an incomplete geometric limit and a comparison of Gromov–Hausdorff and smooth limits. Omitted entirely from an earlier version of this chart.
In a flow on a closed \(3\)-manifold, every point of sufficiently large curvature has a neighborhood that, after rescaling, is close to a corresponding piece of a \(\kappa \)-solution — a neck, a cap, or a closed spherical piece. (Morgan–Tian Chapter 11 supplies the geometric limits of generalized flows this rests on; the appendix, Chapter 19, supplies the neck and cap geometry.)
This converts an analytic classification into a topological description of where and how the manifold is about to pinch, and it is what makes surgery well-defined rather than arbitrary.