1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

APPENDIX C

Glossary

adjoint heat equation. When associated to the Ricci flow, the equa-

tion is

au

at + iJ..u - Ru = 0.
For a fixed metric, the adjoint heat equation is just the backward heat
equation ~~ + iJ..u = 0.
ancient solution. A solution of the Ricci flow which exists on a time
interval of the form (-oo, w) , where w E (-oo, oo]. Limits of dilations about
finite time singular solutions on closed manifolds are ancient solutions which
are 11;-noncollapsed at all scales. For this reason a substantial part of the
.subject of Ricci flow is devoted to the study of ancient solutions.
asymptotic scalar curvature ratio (ASCR). For a complete non-
compact Riemannian manifold,

ASCR (g) = limsup R (x) d (x, 0)^2 ,

d(x,0)-->oo

where 0 E Mn is any choice of origin. This definition is independent of

the choice of 0 EM. For a complete ancient solution of the Ricci flow g (t)

on a noncompact manifold with bounded nonnegative curvature operator,

ASCR (g (t)) is independent of time. The ASCR is used to study the geom-

etry at infinity of solutions of the Ricci flow and in particular to perform

dimension reduction when ASCR = oo.

asymptotic volume ratio (AVR). On a complete noncompact Rie-

mannian with nonnegative Ricci curvature, AVR is the limit of the mono-

tone quantity Vol~;p,r) as r ---+ oo. This definition is independent of the

choice of p E M. Ancient solutions with bounded nonnegative curvature

operator have AVR = 0. Expanding gradient Ricci solitons with positive

Ricci curvatures have AVR > 0.

backward Ricci flow. The equation is

a

arg = 2Rc.

Usually obtained by taking a solution g (t) to the Ricci flow and defining

r (t) ~to - t for some to.

Bernstein-Bando-Shi estimates (also BBS estimates). Short-

time estimates for the derivatives of the curvatures of solutions of the Ricci

flow assuming global pointwise bounds on the curvatures. Roughly, given

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