93 research outputs found
The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics
We consider the class of Levi nondegenerate hypersurfaces in \bC^{n+1}
that admit a local (CR transversal) embedding, near a point , into a
standard nondegenerate hyperquadric in with codimension
small compared to the CR dimension of . We show that, for hypersurfaces
in this class, there is a normal form (which is closely related to the
embedding) such that any local equivalence between two hypersurfaces in normal
form must be an automorphism of the associated tangent hyperquadric. We also
show that if the signature of and that of the standard hyperquadric in
\bC^{N+1} are the same, then the embedding is rigid in the sense that any
other embedding must be the original embedding composed with an automorphism of
the quadric
Two-dimensional shapes and lemniscates
A shape in the plane is an equivalence class of sufficiently smooth Jordan
curves, where two curves are equivalent if one can be obtained from the other
by a translation and a scaling. The fingerprint of a shape is an equivalence of
orientation preserving diffeomorphisms of the unit circle, where two
diffeomorphisms are equivalent if they differ by right composition with an
automorphism of the unit disk. The fingerprint is obtained by composing Riemann
maps onto the interior and exterior of a representative of a shape in a
suitable way. In this paper, we show that there is a one-to-one correspondence
between shapes defined by polynomial lemniscates of degree n and nth roots of
Blaschke products of degree n. The facts that lemniscates approximate all
Jordan curves in the Hausdorff metric and roots of Blaschke products
approximate all orientation preserving diffeomorphisms of the circle in the
C^1-norm suggest that lemniscates and roots of Blaschke products are natural
objects to study in the theory of shapes and their fingerprints
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