386 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS
5. Notes and commentary
§1. The existence and positivity of the heat kernel for a time-dependent
metric was considered by one of the authors [85] (see also [61] and [26]).
For the convenience of the reader we give a proof of this existence in § 1 of
Chapter 23.
For Lemma 26.4 see Theorem 15 on pp. 28-29 of Friedman [61]. For
Lemma 26.1 see (8.4) and (8.5) on p. 27 of [61]. For Corollary 26.15 see for
example Lemma 5.1 of [26].
§2. For Lemma 26.17 see Lemma 5.2 in [26].
For Definition 26.20 see §2 of Grigor'yan [76]. For Lemma 26.21 see
Theorem 2.1 in Grigor'yan [76] and Lemma 2.1 in Chau, Tam, and Yu [26].
Compare also with Lemma 5.1 of Li [117] and Proposition 10.68 in Part II.
For Lemma 26.23 see Lemma 5.3 of [26].
Our exposition of the proof of Theorem 26.32 follows Lemma 5.3 of Li[117]; note that if ..\1 (M, g) > 0, then one can obtain a better estimate.
§3. For the mean value property for harmonic functions, see pp. 25-26
of Evans [58]. The space-time mean value property for solutions to the heat
equation on Euclidean space is due to Pini [158], Fulks [64], and Watson
[185]. A nice exposition of this property is given in Theorem 3 of §2.3.2 in
Evans [58]. For the mean curvature fl.ow, see Ecker [55]; for the Ricci flow,
see Ecker, Topping, and two of the authors [57].
By the heat kernel upper estimates of Carlen, Kusuoka, and Stroock
[24], Carron [25], Cheng, Li, and Yau [38], Davies [51], Li and Yau [121],
Grigor'yan ['76], Nash [137], Varopoulos [183], and others, the class of Rie-
mannian manifolds with backward heat kernel satisfying condition (26.104)
includes the following classes:
(1) manifolds whose Ricci curvatures are bounded from below,
(2) manifolds with an appropriate Sobolev, Nash, logarithmic Sobolev,
or Faber-Krahn-type inequality,
(3) manifolds with an isoperimetric inequality,
( 4) manifolds with bounded geometry.
For (26.107), due to Hamilton [91], see also Malliavin and Stroock [126],
Theorem 10 in §3 of Ecker, Topping, and two of the authors [57], and §2 of
Chapter 16 and §4 of Appendix E both in Part II.
The local monotonicity formula in Theorem 26.44 has additional con-
sequences, including an improved Harnack estimate. The interested reader
may consult Garofalo and Lanconelli [70] for results in this direction.
§4. The work of Tam [1 77] is related to the earlier work of Shi [1 73]
and Lin and Wang [123] and uses some techniques of Greene and Wu [75],
Cheng and Yau [39], Jost and Karcher [102], Schoen and Yau [168], Karp
and Li [106], Li and Yau [121], and Tam and one of the authors [143] and
[144].