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 xvii
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