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

dµ volume form

dµE Euclidean volume form

dσ or dA volume form on boundary or hypersurface

∆, ∆L, ∆d Laplacian, Lichnerowicz Laplacian,

Hodge Laplacian

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