1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

0. GLOSSARY 511

steady gradient Ricci soliton. A gradient Ricci soliton which is

evolving by the pull-back by diffeomorphisms only. In particular, the metrics

at different times are isometric.

strong maximum principle. For the scalar heat equation it says that

a nonnegative solution which is zero at some point (xo, to) vanishes for all

points (x, t) with t :::; to. For the curvature operator under the Ricci fl.ow

Hamilton's strong maximum principle says that given a solution with

nonnegative curvature operator, for positive short time t E (0, <5) the image of

the curvature operator image (Rm (t)) C A^2 T* Mn is invariant under parallel

translation and constant in time. At each point ( x, t) E Mn x ( 0, <5) the image

of the curvature operator is a Lie subalgebra of A^2 T* M~ ~ so ( n).

tensor. A section of some tensor product of the tangent and cotangent

bundles.

Thurston's geometrization conjecture. The conjecture of William

Thurston that any closed 3-manifold admits a decomposition into geometric
pieces. This subsumes the Poincare conjecture.

total scalar curvature. (See Einstein-Hilbert functional.)

trace Harnack estimate. Given a solution of the Ricci fl.ow with

nonnegative curvature operator, the estimate

8R R

0

t + t + 2 (\JR, V) + 2 Re (V, V) 2: 0

holding for any vector V. Taking V = 0, we have gt (tR (x, t)) 2: 0.
Type I, Ila, llb, III singularity. The classification of singular solu-

tions according to the growth or decay rates of the curvatures. For T < oo,

Type I is when

and Type Ila is when

sup (T - t) IRml < oo

Mx[O,T)

sup (T - t) IRml = oo.

Mx[O,T)

For T = oo, Type III is when

and Type IIb is when

sup t IRml < oo

Mx[O,T)

sup t IRml = oo.
Mx[O,T)
Examples of Type I singularities are shrinking spherical space forms and
neckpinches.
Uhlenbeck's trick. In the Ricci fl.ow it is the method of pulling back
tensors to a bundle isomorphic to the tangent bundle with a fixed metric.
This method simplifies the formulas for various evolution equations involving
the Riemann curvature tensor, its derivatives and contractions.
variation formula. Given a variation of a geometric quantity such as
a metric or a submanifold, the corresponding variation for an associated