2,749 research outputs found
Centro-affine hypersurface immersions with parallel cubic form
We consider non-degenerate centro-affine hypersurface immersions in R^n whose
cubic form is parallel with respect to the Levi-Civita connection of the affine
metric. There exists a bijective correspondence between homothetic families of
proper affine hyperspheres with center in the origin and with parallel cubic
form, and K\"ochers conic omega-domains, which are the maximal connected sets
consisting of invertible elements in a real semi-simple Jordan algebra. Every
level surface of the omega function in an omega-domain is an affine complete,
Euclidean complete proper affine hypersphere with parallel cubic form and with
center in the origin. On the other hand, every proper affine hypersphere with
parallel cubic form and with center in the origin can be represented as such a
level surface. We provide a complete classification of proper affine
hyperspheres with parallel cubic form based on the classification of
semi-simple real Jordan algebras. Centro-affine hypersurface immersions with
parallel cubic form are related to the wider class of real unital Jordan
algebras. Every such immersion can be extended to an affine complete one, whose
conic hull is the connected component of the unit element in the set of
invertible elements in a real unital Jordan algebra. Our approach can be used
to study also other classes of hypersurfaces with parallel cubic form.Comment: Fourth version, 35 pages. A missing case has been added to the
classificatio
Boundary behaviour of Weil-Petersson and fiber metrics for Riemann moduli spaces
The Weil-Petersson and Takhtajan-Zograf metrics on the Riemann moduli spaces
of complex structures for an -fold punctured oriented surface of genus
in the stable range are shown here to have complete asymptotic
expansions in terms of Fenchel-Nielsen coordinates at the exceptional divisors
of the Knudsen-Deligne-Mumford compactification. This is accomplished by
finding a full expansion for the hyperbolic metrics on the fibers of the
universal curve as they approach the complete metrics on the nodal curves above
the exceptional divisors and then using a push-forward theorem for conormal
densities. This refines a two-term expansion due to Obitsu-Wolpert for the
conformal factor relative to the model plumbing metric which in turn refined
the bound obtained by Masur. A similar expansion for the Ricci metric is also
obtained
Determinantal representations of hyperbolic plane curves: An elementary approach
If a real symmetric matrix of linear forms is positive definite at some
point, then its determinant is a hyperbolic hypersurface. In 2007, Helton and
Vinnikov proved a converse in three variables, namely that every hyperbolic
plane curve has a definite real symmetric determinantal representation. The
goal of this paper is to give a more concrete proof of a slightly weaker
statement. Here we show that every hyperbolic plane curve has a definite
determinantal representation with Hermitian matrices. We do this by relating
the definiteness of a matrix to the real topology of its minors and extending a
construction of Dixon from 1902. Like Helton and Vinnikov's theorem, this
implies that every hyperbolic region in the plane is defined by a linear matrix
inequality.Comment: 15 pages, 4 figures, minor revision
Livsic-type Determinantal Representations and Hyperbolicity
Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic
real projective hypersurfaces, and their determinantal representations, play a
key role in the emerging field of convex algebraic geometry. In this paper we
consider a natural notion of hyperbolicity for a real subvariety of an arbitrary codimension with respect to a real -dimensional linear subspace and study its basic
properties. We also consider a special kind of determinantal representations
that we call Livsic-type and a nice subclass of these that we call \vr{}. Much
like in the case of hypersurfaces (), the existence of a definite
Hermitian \vr{} Livsic-type determinantal representation implies hyperbolicity.
We show that every curve admits a \vr{} Livsic-type determinantal
representation. Our basic tools are Cauchy kernels for line bundles and the
notion of the Bezoutian for two meromorphic functions on a compact Riemann
surface that we introduce. We then proceed to show that every real curve in
hyperbolic with respect to some real -dimensional linear
subspace admits a definite Hermitian, or even real symmetric, \vr{} Livsic-type
determinantal representation
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