  Regularity of the deformation of the Fermi surface under short-range
interactions is established for all contributions to the RPA self-energy (it is
proven in an accompanying paper that the RPA graphs are the least regular
contributions to the self-energy). Roughly speaking, the graphs contributing to
the RPA self-energy are those constructed by contracting two external legs of a
four-legged graph that consists of a string of bubbles. This regularity is a
necessary ingredient in the proof that renormalization does not change the
model. It turns out that the self--energy is more regular when derivatives are
taken tangentially to the Fermi surface than when they are taken normal to the
Fermi surface. The proofs require a very detailed analysis of the singularities
that occur at those momenta p where the Fermi surface S is tangent to S+p.
Models in which S is not symmetric under the reflection p to -p are included.
