C. GLOSSARY 505eternal solution. A solution of the Ricci flow existing on the time
interval ( -oo, oo). Note that eternal solutions are ancient solutions which
are immortal.
expanding gradient Ricci soliton ( a.k.a. expander). A gradient
Ricci soliton which is evolving by the pull-back by diffeomorphisms and
scalings greater than 1.
Gaussian soliton. Euclidean space as either a shrinking or expanding
soliton.
geometrization conjecture. (See Thurston's geornetrization con-
jecture.)
gradient flow. The evolution of a geometric object in the direction of
steepest ascent of a functional.
gradient Ricci soliton. A Ricci soliton which is flowing along diffeo-
morphisms generated by a gradient vector field.
Gromoll-Meyer theorem. Any complete noncompact Riemannian
manifold with positive sectional curvature is diffeomorphic to Euclidean
space.
Hamilton's entropy for surfaces. The functional E (g) = JM 2 logR·
Rdμ defined for metrics on surfaces with positive curvature.
Hamilton-Ivey estimate. A pointwise estimate for the curvatures of
solutions of the Ricci flow on closed 3-manifolds (with normalized initial
data) which implies that, at a point where there is a sufficiently large (in
magnitude) negative sectional curvature, the largest sectional curvature at
that point is both positive and much larger in magnitude. In dimension
3 the Hamilton-Ivey estimate implies that the singularity models of finite
time singular solutions have nonnegative sectional curvature.
harmonic map. A map between Riemannian manifolds f : (Mn, g) ____,
(Nm, h) satisfying flg,hf = 0, where flg,h is the map Laplacian. (See map
Laplacian.)
harmonic map heat flow. The equation is ~{ = flg,hf.
Harnack estimate (See differential Harnack estimate.)heat equation. For functions on a Riemannian manifold: ~~ = flu.
This equation is the basic analytic model for geometric evolution equations
including the Ricci flow.
heat operator. The operator Zt -fl appearing in the heat equation.
Hodge Laplacian. Acting on differential forms: fld = -( d<5 + <5d).
homogeneous space. A Riemannian manifold (Mn,g) such that for
every x, y EM there is an isometry l: M ____, M with l (x) = y.
Huisken's monotonicity formula. An integral monotonicity formula
for hypersurfaces in Euclidean space evolving by the mean curvature flow
using the fundamental solution to the adjoint heat equation.
immortal solution. A solution of the Ricci flow which exists on a time
interval of the form (a, oo) , where a E [-oo, oo).
isoperimetric estimate. A monotonicity formula for the isoperimetric
ratios of solutions of the Ricci flow. Examples are Hamilton's estimates