Notation and Symbols
Confusion never stops, closing walls and ticking clocks.
From “Clocks” by Coldplay
The following is a list of some of the notation and symbols which we use
in this book.
× multiplication, when a formula does not fit on
one line
covariant derivative
r ‘set gradient’ of the distance function to a point
heat operator

adjoint heat operator
defined to be equal to
· dot product or multiplication
∇∇f Hessian of f
α dual vector field to the 1-form α
ale asymptotically locally Euclidean
Area area of a surface or volume of a hypersurface
ASCR asymptotic scalar curvature ratio
AVR asymptotic volume ratio
B (p, r) ball of radius r centered at p
bounded curvature bounded sectional curvature (for time-dependent
metrics, the bound may depend on time)
CV J tangent cone at V of a convex set J
Rk
const constant
d+
dt
,
d−
dt
,
d+
dt
,
d−
dt
various Dini time derivatives
d distance
dGH Gromov–Hausdorff distance
volume form
dµE Euclidean volume form
or dA volume form on boundary or hypersurface
∆, ∆L, ∆d Laplacian, Lichnerowicz Laplacian,
Hodge Laplacian
xvii
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