Ebenfelt, Peter; Huang, Xiaojun; Zaitsev, Dmitri 'The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics' in American Journal of Mathematics, 127, (1), 2005, pp 169 - 191
American Journal of Mathematics, 127 1
Our main objective in this paper is to study the class of real hypersurfaces M ⊂ Cn+1
which admit holomorphic (or formal) embeddings into the unit sphere (or, more generally,
Levi-nondegenerate hyperquadrics) in CN+1 where the codimension k := N − n is small
compared to n. Such hypersurfaces play an important role e.g. in deformation theory of
singularities where they arise as links of singularities (see e.g. [BM97]). Another source
is complex representations of compact groups, where the orbits are always embeddable
into spheres due to the existence of invariant scalar products. One of our main results
is a complete normal form for hypersurfaces in this class with a rather explicit solution
to the equivalence problem in the following form (Theorem 1.3): Two hypersurfaces in
normal form are locally biholomorphically equivalent if and only if they coincide up to an
automorphism of the associated hyperquadric. Our normal form here is different from the
classical one by Chern–Moser [CM74] (which, on the other hand, is valid for the whole
class of Levi nondegenerate hypersurfaces), where, in order to verify equivalence of two
hypersurfaces, one needs to apply a general automorphism of the associated hyperquadric
to one of the hypersurfaces, possibly loosing its normal form, and then perform an algebraically
complicated procedure of putting the transformed hypersurface back in normal
form. Another advantage of our normal form, comparing with the classical one, is that
it can be directly produced from an embedding into a hyperquadric and hence does not
need any normalization procedure.
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