1547671870-The_Ricci_Flow__Chow

2. TENSOR EQUATIONS 97


3. Weak maximum principles for tensor equations

The maximum principle is extremely robust: it applies to general classes
of second-order parabolic equations and even to some systems, such as the
Ricci fl.ow. The following result is a simple example of the maximum prin-
ciple for systems. Recall that one writes A 2:: 0 for a symmetric 2-tensor A
if the quadratic form induced by A is positive semidefinite.

THEOREM 4.6 (tensor maximum principle, first version: non-negativity
is preserved). Let g (t) be a smooth I -parameter family of Riemannian met-
rics on a closed manifold Mn. Let a(t) E C^00 (T Mn &:is T Mn) be a
symmetric (2, 0)-tensor satisfying the semilinear heat equation
a
at a 2:: ,6.g(t)a + /3,

where f3(a,g,t) is a symmetric (2,0)-tensor which is locally Lipschitz in all

its arguments and satisfies the null eigenvector assumption that

/3 (V, V) (x, t) = (/3ij ViVj) (x, t) 2:: 0

whenever V (x, t) is a null eigenvector of a (t), that is whenever

(aijVJ) (x,t) = 0.

If a (0) 2:: 0 (that is, if a (0) is positive semidefinite), then a (t) 2:: 0 for all

t 2:: 0 such that the solution exists.

This result is in a sense a prototype for more advanced tensor maximum
principles that we shall encounter later. So before giving the full proof, we
will describe the strategy and key concepts behind it.

IDEA OF THE PROOF. Recall that one proves the scalar maximum prin-

ciple (for example, Theorem 4.2) by a purely local argument at a point and

the first time when the solution becomes zero. The tensor maximum prin-

ciple essentially follows from the scalar maximum principle by applying the

tensor to a fixed vector field. To illustrate this, suppose that a > 0 for all

0 ::::; t < to, but that (xo, to) is a point and time and v E Tx 0 Mn is a vector

such that

Then aijWiWj (x, t) 2:: 0 for all x E Mn, t E [O, to], and tangent vectors

W E TxMn. One wants to extend v to a vector field V defined in a space-

time neighborhood of (xo, to) such that V (xo, to) = v and

(4.6a)

( 4.6b)

(4.6c)

av

8t (xo, to)= 0,

V'V (xo, to) = 0,

,6. V (xo, to) = 0.

This may be accomplished by parallel translation (with respect tog (to)) of

v in space along geodesic rays (with respect tog (to)) emanating from xo,