1547671870-The_Ricci_Flow__Chow

3. ADVANCED MAXIMUM PRINCIPLES 101

Let F : £ x [O, T] ---+ £ be a continuous map such that F (.,., t) : £ ---+ £ is
fiber-preserving for each t E [O, T], and F (-, x, t) : Ex ---+ Ex is Lipschitz for
each x E Mn and t E [O, T]. Let K be a closed subset of E. Then we can
state the following result from [59]. Although it is quite general, we still call
it a 'tensor maximum principle' because we shall apply it in the case that
g (t) is a solution of the Ricci flow and E is a tensor bundle.

THEOREM 4.8 (tensor maximum principle, second version: ODE gives
pointwise bounds for PDE). Under the assumptions above, let a (t): 0:::; t:::;
T be a solution of the nonlinear PDE

{) A

ata=.6.a+F(a)

such that a (0) E K. Assume further that:

• K is invariant under parallel translation by V ( t) for all t E [O, T];

and

• Kx ~Kn 7r-l (x) is a closed convex subset of Ex ~ 7r-l (x) for all
  xEMn.

Then if every solution of the ODE

d

dt a= F (a)

a (0) E Kx

defined in each fiber Ex remains in Kx, the solution a (t) of the PDE remains

in K.

There are two important generalizations of this result which will be

useful for us. Both are proved in [33] (though some special cases appear

in Hamilton's work). In the first, one allows the set K to depend on time,

obtaining a time-dependent maximum principle for systems.

THEOREM 4.9 (tensor maximum principle, third version: ODE controls

PDE in time-dependent subsets). Adopt the assumptions above, allowing

K(t) to be a closed subset of E for all t E [O, T ]. Let a(t): 0 St ST

be a solution of the nonlinear PDE

{) A

at a = 6.a + F (a)

such that a (0) EK (0). Assume further that:

• the space-time track UtE[O,T] (K (t) x { t}) is a closed subset of E x
  [O,T];

• K ( t) is invariant under parallel translation by V ( t) for all t E

[O, T); and

• Kx (t) ~ K (t) n 7r-l (x) is a closed convex subset of Ex for all
  x E Mn and t E [O, T].