10 27. NONCOMPACT GRADIENT RICCI SOLITONS
COROLLARY 27.10 (Bounds on f for GRS). Let (Mn,g, f , c) be a complete
normalized G RS and fix 6 E M.
(1) For a steady, for all x EM
(27.41) If (x) - f(O)I::; d(x, 0).
(2) For a shrinker, for all x E M(27.42)
If 0 is a minimum point of f, then(27.43) f(O) S ~and(27.44)1
f(x)::;
4
(d(x,O)+J271)^2.(3) For an expander, for all x EM
(27.45)
1 ( - - 1 /2) 2 n n
- 4
d(x, 0) + (-4f(O) + 2n) + 2 ::; f (x)::; 2·
PROOF. (1) This follows from integrating inequality (27.29) along minimal
geodesics.
(2) Since f ~ 0 , this follows from squaring the right-hand inequality in (27.30).
Now suppose that 0 is a minimum point off. Since 6.f ( 0) ~ 0, by R+6.f = ~
we have R(O)::; ~-Hence, by R+ 1Vfl^2 =fand1Vfl^2 (0) = 0 , we conclude that
f(O)::; ~- Applying this to (27.42) yields (27.44).
(3) Since f ::; ~,this follows from squaring the right-hand inequality in (27.31).
DFor shrinkers, the potential function is in fact uniformly equivalent to the dis-
tance squared. The elegant proof of this relies on the second variation of arc length
formula and integration by parts. Let a+ = max {a, 0} for a E R We have the
following originally due to H .-D. Cao and D. Zhou and later refined by Haslhofer
and Muller (with the sharper constants as presented here).
THEOREM 27.11 (Lower bound for f on shrinkers). Let (Mn,g,f,-l) be a
complete normalized shrinking GRS. Given 6 EM, we have
(27.46) f (x) :;. ~ ( ( d(x, 6) - 2-,/1(6) - 4n +DJ'
If 0 is a minimum point of f, then