1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

7. N OTES AND COMMENTARY 33


8. N ates and commentary
   There are m any works on GRS which we have not discussed in this chapter. We
   have not discussed some important advances in t he contruction of K ahler and non-
   Kiihler Ricci solitons. Some works we would like to mention a re Xu-Jia Wang and
   Xiaohua Zhu [431], Fabio Podesta and Andrea Spiro [329], and Andrew Dance r
   and McKenzie Wang [90]. The results in this ch apter a re due to va rious authors.
   We give some citations b elow.

§1. For Exercise 27.1, see §1.3 of [312].

The proof of Theorem 27.2(1), using the elliptic maximum principle, follows

Theorem 1.3(ii) in [455]. The proof of Theorem 27 .2(2) follows Theorem 0. 5 in

[451]. For a n extension t o a ncient solutions (discussed in the next chapter) , see

[61].

Theorem 27 .4 follows from the formula (27.6) for GRS due to Ha milton com-

bined with Theorem 27 .2. For the shrinking case, see (2.3) in [48] (see also [45]).

Corollary 27 .7 is Theorem 1.3(i) in [455].

For Proposition 27.8, see [326] or [451]. For the claim st at ed in its proof, see

also Proposition 2 in [319].

§2. Rega rding shrinkers , (27. 4 2) is (2 .2) in [48].

For Theorem 27.11, see Theorem 1.1 in [48] and the refinement in [146]. If the

Ricc i curvature of (Mn, g) is bounded , then this estimat e is in [312].

For Corollary 27. 16 , see [105].

Also relat ed is the work in [105], where the properness off is proven and from

which a nonsha rp form of the quadratic growth of f may b e derived.

Theorem 27.19 is in [268].

Theorem 27.24 and Corollary 27.25 a re in [95].

§3. For Theorem 27. 26 , see [291] and [78].
See [441] and [264] for an estima t e for the potential functions of steady GRS.
See [32], [46], [132], [138], and [264] for furt her works on the qualitative
as p ect s of steady GRS.

§4. For (27. 8 1), see (3. 7) of [48].
The result that shrinkers have at most Euclidean volume growth, in Theorem
27.33 and Theorem 27 .42, is prima rily due to [48], with a t echnical hypothesis they
fir st assumed removed in [263]. That the lim sup of the volume ratios , as the radius
t ends to infinity, is bounded a bove by a const a nt dep ending only on dimension is
in [146].
The p art of Theorem 27 .33 that the AVR of a shrinker exists is in [79]. Their
work built upon the eariler works in [53], [45], [451], [64], and [290]. Some work
relat ed t o the above is in [105] and is by Ha milton (see Proposition 9.46 in [77]).
For Proposition 27.35, see [79].

Corollary 27.36 is in [451]. It extends the earlier results in [48] and [263]. The

alterna t e proof we give is due to Bo Yang.
For Theorem 27.41, see Theorem 1.2(a) in [434] for example. See also [442].
For some earlier related work, see [202] and [16].
The proof of Theorem 27.42 t hat we present, using the Riccati equa tion, is in
[267].
For further work on GRS, see [453].