If the cross-section 
of a tube does not grow exponentially
in one direction or the other,
there is no hope that 
will be positive.
The reason is that
measures the exponential rate of decay
of the heat kernel for the tube,
and since the flow of heat along the tube is slow and no heat is lost,
the temperature can't decay exponentially unless
there is an exponential amount of material over which
to distribute the heat.
On the other hand,
if is growing exponentially
with a certain minimum rate,
and if it doesn't waver too much,
then we can get a positive lower bound for .
The bigger the minimum growth rate and the smaller the amount of
wavering,
the better the lower bound will be.
Lemma. Suppose that
where
is piecewise 
and
Then
Proof. Let
Then
so
Definition.
If 
is measurable and satisfies
for some piecewise function
with
then we say that f grows more or less exponentially, with rate a and factor K.
Proposition.
Suppose for a cutting of M into tubes we can find
constants a, K
such that for almost all 
,
grows more or less exponentially,
with rate a and factor K.
Then
