502 C. GLOSSARY
a solution of the Ricci fl.ow, if !Rm (t) lg(t) :S K on a time interval [O, T) of
length on the order of K-^1 , then
l\7m Rm ( t) lg(t) :S ~::%on that interval. The BBS estimates are used to obtain higher derivative of
curvature estimates from pointwise bounds on the curvatures. In particular,
they are used in the proof of the C^00 compactness theorem for sequences
of solutions assuming only uniform pointwise bounds on the curvatures and
injectivity radius estimates.
Bianchi identity. The first and second Bianchi identities are
~jke + Rjkie + Rkij£ = 0,
\7iRjk£m + 'iljRki£m + \7kRij£m = 0,respectively. The Bianchi identities reflect the diffeomorphism invariance of
the curvature. In the Ricci fl.ow they are used to derive various evolution
equations including the heat-like equation for the Riemann curvature tensor.
Bishop-Gromov volume comparison theorem. An upper bound
for the volume of balls given a lower bound for the Ricci curvature. This
bound is sharp in the sense that equality holds for complete, simply-connected
manifolds with constant sectional curvature.
Bochner formula. A class of formulas where one computes the Lapla-
cian of some quantity (such as a gradient quantity). Such formulas are often
used to prove the nonexistence of nontrivial solutions to certain equations.
For example, harmonic 1-forms on closed manifolds with negative Ricci cur-
vatures are trivial. In the Ricci fl.ow, Bochner-type formulas (where the
Laplacian is replaced by the heat operator) take the form of evolution equa-
tions which yield estimates after the application of the maximum principle.
bounded geometry. A sequence or family of Riemannian manifolds
has bounded geometry if the curvatures and their derivatives are uniformly
bounded (depending on the number of derivatives).
breather solution. A solution of the Ricci fl.ow which, in the space of
metrics modulo diffeomorphisms, is a periodic orbit.
Bryant soliton. The complete, rotationally symmetric steady gradi-
ent Ricci soliton on Euclidean 3-space. The Bryant soliton has sectional
curvatures decaying inverse linearly in the distance to the origin and hence
has ASCR = oo and AVR = 0 .. It is also expected to be the limit of the
conjectured degenerate neckpinch.
Buscher duality. A duality transformation of gradient Ricci solitons
on warped products with circle or torus fibers.
Calabi's trick. A typical way to localize maximum principle arguments
is to multiply the quantity being estimated by a cut-off function depending
on the distance to a point. Calabi's trick is a way to deal with the issue
of the cut-off function being only Lipschitz continuous (~ince the distance
function is only Lipschitz continuous).