11 Ricci flow with surgery
Here the object being studied changes character: the manifold itself is cut and reglued as the flow proceeds. The encouraging discovery on reading Morgan–Tian’s structure is that they do not treat this as an informal procedure. The flow is redefined on a single geometric object — a surgery space-time — and the surgery process becomes an equation on that object. That is precisely the kind of reformulation formalization needs, and it moderates an earlier claim here that surgery is unformalizable in principle. It remains unattempted anywhere, and it remains long.
The explicit flow on \(\mathbb {R}^3\) with rotationally symmetric initial metric, asymptotic to a cylinder, used as the model for the cap glued in at surgery. (Morgan–Tian Chapter 12, seven sections.)
Carries its own PDE prerequisite. Uniqueness of the standard solution is proved through the harmonic map flow (12.5), a second geometric flow with its own existence theory. This road needs more than one parabolic theory.
The metric on a sufficiently deep neck may be replaced by a capped metric with controlled curvature, preserving the pinching hypotheses. (Morgan–Tian Chapter 13.)
A space-time whose time-slices are the evolving manifolds, singular along the surgery caps, carrying a horizontal metric satisfying the generalized Ricci flow equation. (Morgan–Tian Chapter 14.)
Let \((M, g_0)\) be a closed Riemannian \(3\)-manifold containing no embedded, locally separating \(\mathbb {R}P^2\). Then there is a Ricci flow with surgery defined for all \(t \in [0,\infty )\) with initial metric \(g_0\), whose discontinuity times form a discrete subset of \([0,\infty )\), and where the topological change across a surgery time is a connected sum decomposition together with removal of components diffeomorphic to \(S^2 \times S^1\), \(\mathbb {R}P^3 \# \mathbb {R}P^3\), the non-orientable \(S^2\)-bundle over \(S^1\), or a manifold of constant positive curvature. (Morgan–Tian Theorem 0.3; Chapters 15–17.)
The \(\mathbb {R}P^2\) hypothesis is not decoration and was missing from an earlier version of this chart. The proof is a mutual induction over surgery times: parameters are chosen in advance, and non-collapsing (Chapter 16) and the canonical neighborhood assumption (Chapter 17) are re-established after every surgery. Chapters 15–17 are the bulk of the book.