1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

382 8. APPLICATIONS OF THE REDUCED DISTANCE

EXERCISE 8.1. Show that if ( JVin, g) is a complete noncom pact Rie-

mannian manifold with Reg 2:: -K for some K E IR, then the integral

defining V (g, r) converges for all T > 0.

In Euclidean space V is the integral of the heat kernel, which is the

constant 1. Note also that for any (JVin,g) and p EM,

(8.2) T-->0 lim V (g, r) = 1

essentially since manifolds are locally Euclidean.

EXERCISE 8.2. Prove (8.2).

Let

u (x, r) ~ (47rr)-n/2 e-d(x,p)2/4T,

which is a Lipschitz function, and let d (x) ~ d (x,p). We can think of

v (g, r) = r u (x, r) dμ (x) as a weighted volume centered at p with the

};Vt

radial weight function u. As T --t 0, the weight u concentrates at p and as

T --too, u diffuses throughout M.

REMARK 8.3. If Mis closed, then the upper bound

V(g,r) :S JM (47rr)-n/^2 dμ(x) = (47rr)-n/^2 Vol(g)

implies that lim 7 __, 00 V (g, r) = 0.

Now assume that (JVin, g) is complete with nonnegative Ricci curvature.

Since Reg 2:: 0, the Bishop-Gromov volume comparison theorem says that

the volume ratio r-n Vol B (p, r) is a nonincreasing function of r. It is thus

natural to expect that V (g, T) is a nonincreasing function of T since as r

increases, the weighting favors larger radii. Indeed we have

LEMMA 8.4 (Static reduced volume monotonicity). If ( JVin, g) is com-

plete with Reg 2:: 0, then

(8.3) d~V(§,r)= JM (!-t::.)udμ:SO.

In particular, by (8.2),
V(g,r) :S 1 for all T > 0.
REMARK 8.5. Clearly this lemma implies lim 7 __, 00 V (g, r) E [O, 1] exists.
PROOF. We compute that u is a subsolution, in the weak sense, to the
heat equation:

(

(^8) ) _ ( n d2 ( dt::.d + IY'dl
2
) d2 2 )

• 
  
  
  
  • t::. u - u --+ - + - - IV'dl
    ~ ~ ~ ~ ~
  
  
  
  

(8.4) :S 0,