508 C. GLOSSARY
monotonicity formula. Any formula which implies the monotonicity
of a pointwise or integral quantity under a geometric evolution equation.
Examples are entropy and Harnack estimates.
μ-invariant. An invariant of a metric and a positive number (scale)
obtained from Perelman's entropy functional W (g,f, T) by taking the infi-
mum over all f satisfying the constraint fMn (4nT)-nl^2 e-f dμ = 1. There is
a monotonicity formula for this invariant under the Ricci flow.
neckpinch. A finite time (Type I) singular solution of the Ricci flow
where a region of the manifold asymptotically approaches a shrinking round
cylinder sn-l x R Sufficient conditions for initial metrics on sn for a neck-
pinch have been obtained by Angenent and one of the authors.
no local collapsing. (Also abbreviated NLC.) A fundamental theorem
of Perelman which applies to all finite time solutions of the Ricci flow on
closed manifolds. It says that given such a solution of the Ricci flow and a
finite scale p > 0, there exists a constant r;, > 0 such that for any ball of
radius r < p with curvature bounded by r-^2 in the ball, the volume ratio of
the ball is at least r;,. We say that the solution is r;,-noncollapsed below
the scale p.
v-invariant. A metric invariant obtained from the μ-invariant by taking
the infimum over all T > 0. This invariant may be -oo.
null-eigenvector assumption. A condition, in the statement of the
maximum principle for 2-tensors, on the form of a heat-type equation which
ensures that the nonnegativity of the 2-tensor is preserved under this heat-
type equation.
parabolic equation. In the context of Ricci flow, a heat-type equa-
tion (which is second-order). In general, parabolicity of a nonlinear partial
differential equation is defined using the symbol of its linearization.
Perelman's energy. The functional
This invariant appeared previously in mathematical physics (e.g., string the-
ory) and f is known as the dilaton.
Perelman's entropy. The following functional of the triple of a metric,
a function, and a positive constant:
W(g, f, T) =!Mn (T ( R + j\7 f1^2 ) + (f - n)) (4nT)-n/^2 e-f dμ.
Perelman's Harnack (LYH) estimate. A differential Harnack (e.g.,
gradient) estimate for fundamental solutions of the adjoint heat equation
coupled to the Ricci flow.
Perelman solution. The non-explicit 3-dimensional analogue of the
Rosenau solution. The Perelman solution is rotationally symmetric and has
positive sectional curvature. Its backward limit as t -+ -oo is the Bryant
soliton.