- HEAT BALLS AND THE SPACE-TIME MEAN VALUE PROPERTY 375
3.4. Mean value property under Ricci flow.
The mean value property can also be extended, as an inequality, to
the case where the Riemannian metric is evolving by the Ricci fl.ow for
supersolutions to the associated heat equation under the assumption that
the scalar curvature of the evolving metric is nonnegative. For this we invoke
the concept of the reduced distance function (see Chapter 7 of Part I).
Let (Mn,g(t)), t E [O,T], be a solution to the Ricci fl.ow. Let £(y,T)
denote the reduced distance function, defined with respect to some basepoint
(xo, to) EM x (0, T] and where T ~to - t. Let(26.118)For any r > 0, one defines the heat ball using the reduced distance
function by
(26.119) Er~ {(y, t) : v(y, T (t)) 2: r-n and t::; to}.
Using standard estimates for the reduced distance, one can check that Er is
compact for r sufficiently small.
Analogous to (26.90) and (26.108), let
7/Jr(Y, t) ~ logv(y, T (t)) + n logr
and for any smooth function u(x, t) let
(26.120) P(r) ~ j j (IV7/Jrl^2 + 7/JrR) udμg(t)dt
(compare with (26.98) and (26.110)). Note that 'I/Jr 2: 0 in Er.
The following monotonicity formula was proved in [57].THEOREM 26.44. Let (Mn, g(t)), t E [O, T], be a complete solution to
the Ricci flow and let
I(r) ~ P~).
rThen for any smooth function u and 0 < ri < rz,
(26.121)
J(r2) - J(r1)= -1:
2
r;l lr (; ( :t + ~ - R) V +'I/Jr ( :t - ~) U) dμg(t)dt dr.Recall that Perelman proved that (see (7.91) in Part I)
8£ 2 n
OT - ~£ + IV£1 - R + 2T 2: 0,that is,
( -+~-R^8 ) v= ( --~£+1V£1 8£^2 -R+-n ) v2:0.