Chapter 8. Applications of the Reduced Distance.
Now ... the basic principle of modern mathematics is to achieve a complete fusion [of]
'geometric' and 'analytic' ideas. -Jean DieudonneIn this chapter we give some geometric applications of the reduced dis-
tance for Ricci fl.ow. We give two proofs of the monotonicity of the reduced
volume. This result is the Ricci fl.ow analogue of the Bishop-Gromov volume
comparison theorem in Riemannian geometry. A beautiful and striking as-
pect of this monotonicity formula is that unlike Hamilton's matrix Harnack
inequality and most other monotonicity formulas, no curvature assumption
is needed. Using the reduced volume monotonicity, we prove a weakened
no local collapsing theorem and we also prove that certain backward limits
of ancient 11;-solutions are gradient shrinkers. All of these results are due to
Perelman.
Throughout this chapter we shall use· (Mn, g) to denote a complete
Riemannian manifold and (Mn, g ( T)) to denote a solution of the backward
Ricci fl.ow. ·
1. Reduced volume of a static metric
We begin by defining the reduced volume of a static metric since this is
technically easier than the Ricci fl.ow case yet it still exhibits many of the
ideas.
1.1. The reduced volume for a static metric and its mono-
tonicity when Re 2: O. Consider the following functional for a complete
Riemannian manifold (Mn, g). Given a point p E M, define the static
reduced volume by
(8.1) V (g, T) ~JM (47rT)-n/^2 e-d(x,p)
2
/4T dμ (x).This geometric invariant depends on g, T and p. Clearly V is positive. In
general, we can think of this as the integral of the Euclidean heat kernel
transplanted to a manifold via the exponential map. If the Ricci curvatures
of (M, g) are bounded below by a constant, then by the Bishop-Gromov
volume comparison theorem, which gives an upper bound for the volumes
of balls, the integral defining V (g, T) converges for all T > 0 even when M
is noncompact.
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