C. GLOSSARY 507little loop conjecture. Hamilton's conjecture which is essentially
equivalent to Perelman's no local collapsing theorem (e.g., the conjecture
is now a theorem).Li-Yau-Hamilton (LYH) inequality. (See differential Harnack
estimate.)
locally homogeneous space. (See also homogeneous space.) A
Riemannian manifold (Mn, g) such that for every x, y E M there exist open
neighborhoods U of x and V of y and an isometry i: U ----t V with i (x) = y.
A complete simply-connected locally homogeneous space is a homogeneous
space.
locally Lipschitz function. A function which locally has a finite Lip-
schitz constant.
logarithmic Sobolev inequality. A Sobolev-type inequality which
essentially bounds the classical entropy of a function by the L^2 -norm of the
first derivative of the function.
long existing solutions. Solutions which exist on a time interval of
infinite duration.
long-time existence. The existence of a solution of the Ricci fl.ow on a
closed manifold as long as the Riemann curvature tensor remains bounded.
By a result of Sesum, in the above statement the Riemann curvature tensor
may be replaced by the Ricci tensor.
map Laplacian. Given a map f : (Mn, g) ----t (Nm, h) between Rie-
mannian manifolds, b.. 9 ,hf is the trace with respect to g of the second co-
variant derivative off. (See (3.39) in Volume One.)
matrix Harnack estimate. In Ricci fl.ow a certain tensor inequality
of Hamilton for solutions with bounded nonnegative curvature operator. A
consequence is the trace Harnack estimate. One application of the matrix
Harnack estimate is in the proof that an ancient solution with nonnegative
curvature operator and which attains the space-time maximum of the scalar
curvature is a steady gradient Ricci soliton.
maximum principle. (Also called the weak maximum principle.)
The first and second derivative tests applied to heat-type equations to obtain
bounds for their solutions. The basic idea is to use the inequalities b..u 2: 0
and \7 u = 0 at a spatial minimum of a function u and the inequality ~~ :S 0
at a minimum in space-time up to that time. This principle applies to
scalars, tensors, and systems.
maximum volume growth. A noncompact manifold has maximum
volume growth if for some point p there exists c > 0 such that Vol B (p, r) 2:
crn for all r > 0. This is the same as AVR > 0 (when Re 2: 0).
mean curvature flow. The evolution of a submanifold in a Riemannian
manifold in the direction of its mean curvature vector.
modified Ricci flow. An equation of the form gtg = -2 (Rc+'V'Vf),
where f is a function on space-time. Often coupled to an equation for f
such as Perelman's equation ~{ = -R - b..f.