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In the previous lecture, we showed that every -solution generated at least one asymptotic gradient shrinking soliton . This soliton is known to have the following properties:
- It is ancient: t ranges over .
- It is a Ricci flow.
- M is complete and connected.
- The Riemann curvature is non-negative (though it could theoretically be unbounded).
- is non-negative.
- M is -noncollapsed.
- M is not flat.
- It obeys the gradient shrinking soliton equation
for some smooth f.
The main result of this lecture is to classify all such solutions in low dimension:
Theorem 1. (Classification of asymptotic gradient shrinking solitons) Let be as above, and suppose that the dimension d is at most 3. Then one of the following is true (up to isometry and rescaling):
- d=2,3 and M is a round shrinking spherical space form (i.e. a round shrinking , , , or for some finite group acting freely on ).
- d=3 and M is the round shrinking cylinder or the oriented or unoriented quotient of this cylinder by an involution.
The case d=2 of this theorem is due to Hamilton; the compact d=3 case is due to Ivey; and the full d=3 case was sketched out by Perelman. In higher dimension, partial results towards the full classification (and also relaxing many of the hypotheses 1-8) have been established by Petersen-Wylie, by Ni-Wallach, and by Naber; these papers also give alternate proofs of Perelman’s classification.
To prove this theorem, we induct on dimension. In 1 dimension, all manifolds are flat and so the claim is trivial. We will thus take d=2 or d=3, and assume that the result has already been established for dimension d-1. We will then split into several cases:
- Case 1: Ricci curvature has a zero eigenvector at some point. In this case we can use Hamilton’s splitting theorem to reduce the dimension by one, at which point we can use the induction hypothesis.
- Case 2: Manifold noncompact, and Ricci curvature is positive and unbounded. In this case we can take a further geometric limit (using some Toponogov theory on the asymptotics of rays in a positively curved manifold) which is a round cylinder (or quotient thereof), and also a gradient steady soliton. One can easily rule out such an object by studying the potential function of that soliton on a closed loop.
- Case 3: Manifold noncompact, and Ricci curvature is positive and bounded. Here we shall follow the gradient curves of f using some identities arising from the gradient shrinking soliton equation to get a contradiction.
- Case 4: Manifold compact, and curvature positive. Here we shall use Hamilton’s rounding theorem to show that one is a round shrinking sphere or spherical space form.