multiple time scale dynamics with two fast variables and one slow ...

from the evolution equations hold and then extend this procedure by com-
pactness of B.
• Step 5: This step provides the basic estimates to control the evolution of
all 2-forms inside B. The estimates appear to be rather involved but the
essential tool is just Gronwall’s Lemma.
• Step 6: This step collects all the previous results and derives from estimates
at q the final estimates at ¯q. This final part of the proof is subdivided into
three parts. The first tracks M from q to Sǫ, the second follows M near Sǫ
and the third deals with the exit from Sǫ towards ¯q. The overall patching
strategy is the same as in Step 4.
Step 1: We start with a remark on 2-forms in R 4 . Letσi for i=1, 2, 3, 4 be
coordinates in R 4 . Observe that if P is a 2-plane then the natural first attempt
to define how to evaluate a 2-form on P is to take two independent vectors
that span P and then evaluate the 2-form on those vectors; this process is only
defined up to a multiple as we can scale basis vectors inside P. To avoid this
ambiguity consider a unit rectangle R of P and evaluate the form on the vectors
spanning it. We can interpret this process geometrically in the usual way; for
example, consider (dσ1∧ dσ2)(P) which can be evaluated projecting R onto the
(σ1,σ2)-plane along the coordinate axesσ3 andσ4 and then taking the area of
this projection. Hence we have defined the evaluation of 2-forms on planes up
to orientation; the orientation will not matter for us in the following as we will
only consider absolute values when evaluating 2-forms on planes. Furthermore
observe that e.g. the (σ1,σ2)-plane is characterized by vanishing of all basic 2-
forms execept dσ1∧ dσ2 and dσ2∧ dσ1.
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