1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

504 0. GLOSSARY

cosmological constant. A constant c introduced into the Ricci flow

equation:

a

at9ij = -2 (Rij + C9ij).

The case c = ~ is useful in converting expanding Ricci solitons to steady

Ricci solitons.

cross curvature flow. A fully nonlinear flow of metrics on 3-manifolds

with either negative sectional curvature everywhere or positive sectional

curvature everywhere.

curvature gap estimate. For long existing solutions, a time-dependent

lower bound for the spatial supremums of the curvatures. See Lemmas 8.7,

8.9, and 8.11 in [111].

curvature operator. The self-adjoint fiberwise-linear map

Rm: A^2 Mn---+ A^2 Mn

defined by Rm (a)ij ~ RijkJ!.D'.Ji.k·

curve shortening flow (CSF). The evolution equation for a plane

curve given by

ax

• = -K,V
  at '
  where K, is the curvature and v is the unit outward normal. It is useful to
  compare the CSF with the Ricci flow (especially on surfaces).
  degenerate neckpinch. A conjectured Type Ila singularity on the n-
  sphere where a neck pinches at the same time its cap shrinks leading to a
  cusp-like singularity. Such a singularity has been proven by Angenent and
  Velazquez to occur for the mean curvature flow.
  DeTurck's trick. (See also Ricci-DeTurck flow.) A method to prove
  short-time existence of the Ricci flow using the Ricci-DeTurck flow.
  differential Harnack estimate. Any of a class of gradient-like esti-
  mates for solutions of parabolic and heat-type equations.
  dilaton. (See Perelman's energy.)
  dimension reduction. For certain classes of complete, noncompact
  solutions of the Ricci flow, a method of picking points tending to spatial
  infinity and blowing down the corresponding pointed sequence of solutions
  to obtain a limit solution which splits off a line.
  Einstein-Hilbert functional. The functional of Riemannian metrics:

E (g) = fMn Rdμ, where R is the scalar curvature.

Einstein metric. A metric with constant Ricci curvature.
Einstein summation convention. The convention in tensor calculus
where repeated indices are summed. Strictly speaking the summed indices
should be one lower and one upper, but in practice we do not always bother
to lower and raise indices.
energy. (See Perelman's energy.)
entropy. (See classical entropy, Hamilton's entropy for surfaces,
and Perelman's entropy.)