Ian Agol's Research Blog
3/1/03
Hamilton's paper, "Four-manifolds with Positive Isotropic
Curvature" extends his analysis of 3-manifolds with positive Ricci curvature,
and 4-manifolds with positive curvature operator. In two and 3-dimensions,
positive sectional curvature is preserved by the Ricci flow. But in four
or more dimensions, it is believed not to be preserved. I believe this
has something to do with the fact that the sectional curvatures do not
determine the full curvature tensor in >3 dimensions. Anyway, Hamilton
shows that for 4-manifolds, positive isotropic curvature is preserved by
the Ricci flow. I don't have an intuitive feel for positive isotropic curvature,
which seems to be somewhere in between positive Ricci curvature and positive
curvature operator, so I won't try to describe it. But I've skimmed through
the paper, since the method Hamilton uses to cut off neck-pinch singularities
is the same method he wishes to use for 3-manifolds. Anyway, Hamilton gives
a classification of 4-manifolds with positive isotropic curvature, by evolving
under Ricci flow. If the curvature blows up in finite time, then he shows
that the manifold gets pinched along a neck, which looks like a shrinking
3-dim. spherical space form x I in the limit. Hamilton proves that if this
spherical space form is not S^3 or RP^2 with twisted normal bundle, then
it is pi_1-injective (the pi_1-injectivity is a nice argument making use
of earlier work of Micallef and Moore, and can be read independently of
the rest of the paper). Otherwise, he can surger near the neck and continue
the flow, until after finitely many surgeries one obtains manifolds with
positive curvature operator, in which case they must be spherical space
forms, by Hamilton's earlier result. Then he reverses the surgeries to
conclude that the initial manifold was a connect sum of space forms. Thus,
he doesn't get a full classification, since he must assume that the initial
manifold has no pi_1-injective 3-space forms. I think it might be possible
to finish Hamilton's classification using an orbifold trick. If the neck
pinching occurs along a pi_1-injective S^3/G, then do surgery to obtain
an orbifold with positive isotropic curvature with two orbifold points
which have neighborhoods of the form B^4/G. Then one would continue the
flow as Hamilton does. Hamilton shows in his paper that if one takes quotients
S^3xS^1 or S^4, then these are conformally flat with positive scalar curvature,
and one can perform connect sums of these preserving both properties, which
in turn implies that the resulting manifold has positive isotropic curvature.
So I reckon that 4-orbifolds with positive isotropic curvature should be
connect sums of orbifold quotients of S^3xS^1 or S^4. Doing connect
sums along orbifold points should correspond to connect sum with S^3/G
x S^1, since one should only have such orbifold points if an irreducible
factor of the orbifold was already of the form S^4/G, where G acts by suspension.
I haven't gotten around to checking whether all of Hamilton's argument
generalizes to the orbifold case, but it seems promising.