506 C. GLOSSARYfor solutions on closed surfaces and Type I singular solutions on closed 3-
manifolds.
Jacobi field. A variation vector field of a 1-parameter family of geodesics.
Kahler-Ricci flow. The Ricci fl.ow of Kahler metrics. Note that on a
closed manifold, an initial metric which is Kahler remains Kahler under the
Ricci fl.ow.11;-noncollapsed at all scales. A metric (or solution) which is K-
noncollapsed below scale p for all p < oo.
11;-noncollapsed below the scale p. A Riemannian manifold satisfying
Vol~~x,r) 2: /1; for any metric ball B(x,r) with I Rm I :S r-^2 in B(x,r) and
r < p.
11;-solution (or ancient 11;-solution). A complete ancient solution which
is K-noncollapsed on all scales, has bounded nonnegative curvature operator,
and is not fl.at. In dimension 3, a large part of singularity analysis in Ricci
flow is to classify ancient K-solutions.
£-distance. A space-time distance-like function for solutions of the
backward Ricci flow obtained by taking the infimum of the £-length. The
£-distance between two points may not always be nonnegative.
£-distance function. (See reduced distance.)
£-exponential map. The Ricci flow analogue of the Riemannian ex-
ponential map.
£-geodesic. A time-parametrized path in a solution of the backward
Ricci fl.ow which is a critical point of the £-length functional.
£-Jacobi field. A variation vector field of a 1-parameter family of
£-geodesics.
£-length. A length-like functional for time-parametrized paths in so-
lutions of the backward Ricci fl.ow. The £-length of a path may not be
positive.
Laplacian (or rough Laplacian). On Euclidean space the operator
Ll = 2:?= 1 a(~~) 2 • On a Riemannian manifold, the second-order linear differ-ential operator Ll = ij'\li\lj acting on tensors.
Levi-Civita connection. The unique linear torsion-free connection on
the tangent bundle compatible with the metric. (Also called the Riemannian
connection.)
Lichnerowicz Laplacian. The second-order differential operator LlL
acting on symmetric 2-tensors defined by (Vl-3.6), i.e.,
LlLVij =i= Llvjk + 2gqp R~jk Vrp - gqp Rjp Vqk - gqp Rkp Vjq·
Lichnerowicz Laplacian heat equation. The heat-like equation
gtvij = (LlLv)ij for symmetric 2-tensors.
linear trace Harnack estimate. A differential Harnack estimate for
nonnegative solutions of the Lichnerowicz Laplacian heat equation coupled
to a solution of the Ricci fl.ow with nonnegative curvature operator, which