384 8. APPLICATIONS OF THE REDUCED DISTANCEThat is,
82()()2 ( ()2 2 ) = 1 - 27r.^0 7r•
The square torus is also a nice concrete example for which we can com-
pute the static reduced volume explicitly.EXAMPLE 8.8 (Static reduced volume for square torus). Consider the
torus M_n ~ JRn / (2Zt with the standard fl.at metric g = dxi + · · · + dx;. A
fundamental domain for the covering JRn --+ M is D ~ (-1, l]n. Let p = 0
be the origin so that d (x,p) = !xi for x E (-1, l]n = M. The static reduced
volume is^1
(8.6)using the change of variables x = 2 .J:r. From (8.6) it is clear that V (g, r) is
a decreasing function of r.1.2. Static reduced volume and volume ratios. We may think ofVas the static manifold analogue of Perelman's reduced volume for the
Ricci fl.ow (see (8.16) defined later in this chapter), which is defined similarly
with d^2 /4r replaced by the reduced distance function f. The monotonicity
formula (8.3) is analogous to Perelman's monotonicity of the reduced volume
(8.28). In the Ricci fl.at case, V is the same as Perelman's reduced volume(see Exercise 7.12 and (8.16)). Intuitively the static reduced volume V says
something about volume ratios r-n Vol B (p, r) at scales r rv y!T. Motivated
by these elementary yet a posteriori considerations, we now relate V to the
volume ratio r-n Vol B (p, r) under the assumption that the Ricci curvatures
of g are nonnegative.
Let (Nin, g) be a complete Riemannian manifold with Rey > 0. We
divide the integral V into two parts:
(8.7) V(§,r) = { udμ+ f, udμ.
j B(p,r) j M-B(p,r)For the first term on the RHS, just using the obvious fact that e-r^2 /^47 ::; 1,
we have
[ udμ::; (47rr)-n/^2 VolB (p, r).
}B(p,r)(^1) Since the torus is (Ricci) fl.at, the static metric reduced volume is the same as the
Ricci fl.ow reduced volume of Perelman.