NOTATION AND SYMBOLS

div

E

n

g(X,Y) = (X,Y)

9(t)

h

H

HVJ for V e dj

Hess/

id

IVP

L

LHS

log

X

J

£

C

MCF

9Jlet

MVP

X

nun

NRF

V

un

ODE

PDE

PIC

RF

RHS

i?, Rc, Rm

Rm #

R

Rc(R)

divergence

W1

with the flat Euclidean metric

Christoffel symbols

metric or inner product

time-dependent metric, e.g., solution of

the Ricci flow

second fundamental form

mean curvature

set of closed half-spaces H containing

J c R

f e

with V e OH

Hessian of / (same as VV/)

identity

initial-value problem

length

left-hand side

natural logarithm

a time interval for the Ricci flow

a time interval for the backward Ricci flow

reduced distance or ^-function

Lie derivative or £-length

mean curvature flow

space of Riemannian metrics on a manifold

mean value property

multiplication, when a formula does not fit on

one line

volume of the unit Euclidean (n — l)-sphere

normalized Ricci flow

unit outward normal

volume of the unit Euclidean n-ball

ordinary differential equation

partial differential equation

positive isotropic curvature

Ricci flow

right-hand side

scalar, Ricci, and Riemann curvature tensors

the quadratic Rm # Rm

algebraic curvature operator

a trace of R (of two indices)