3. Basic singularity theory for Ricci fl.ow
- Basic singularity theory for Ricci fl.ow
The knowledge of which geometry aims is the knowledge of the eternal.- Plato
Geometry is knowledge of the eternally existent. -PythagorasAnd perhaps, posterity will thank me for having shown it that the ancients
did not know everything. -Pierre Fermat465In this section we review some basic singularity theory as developed by
Hamilton and discussed in Volume One.3.1. Long-existing solutions and singularity types. For the fol-
lowing, see pp. 234-236 in Section 1 of Chapter 8 in Volume One.
DEFINITION A.34.- An ancient solution is a solution that exists on a past time in-
terval (-oo, w ). - An immortal solution is a solution that exists on a future time
interval (a, oo). - An eternal solution is a solution that exists for all time ( -oo, oo).
DEFINITION A.35 (Singularity types). Let (Mn, g (t)) be a solution of
the Ricci fl.ow that exists up to a maximal time T::::; oo.
• One says (M,g (t)) forms a Type I singularity if T < oo and
sup (T-t) !Rm(·, t)I < oo.
Mx[O,T)• One says (M,g (t)) forms a Type Ila singularity if T < oo and
sup (T-t)!Rm(·,t)l=oo.
Mx[O,T)• One says ( M, g ( t)) forms a Type Ilb singularity if T = oo and
sup t !Rm(·, t)I = oo.
Mx[O,oo)• One says ( M, g ( t)) forms a Type III singularity if T = oo and
sup t I Rm(-, t) I < oo.
Mx[O,oo)
To this we add the following.DEFINITION A.36 (More singularity types). If g (t) is defined on (0, T],
where T < oo, then
• one says ( M, g ( t)) forms a Type Ile singularity as t ---* 0 if
sup t !Rm(·, t)I = oo;
Mx(O,T]