6,724 research outputs found
Eigenfunctions of the Laplacian and associated Ruelle operator
Let be a co-compact Fuchsian group of isometries on the Poincar\'e
disk \DD and the corresponding hyperbolic Laplace operator. Any
smooth eigenfunction of , equivariant by with real
eigenvalue , where , admits an integral
representation by a distribution \dd_{f,s} (the Helgason distribution) which
is equivariant by and supported at infinity \partial\DD=\SS^1. The
geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension
over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the
so-called Bowen-Series transformation. Let be the complex Ruelle
transfer operator associated to the jacobian . M. Pollicott showed
that \dd_{f,s} is an eigenfunction of the dual operator for the
eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic
eigenfunction of for the eigenvalue 1, given by an
integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}}
\dd_{f,s} (d\eta), \noindent where is a -valued
piecewise constant function whose definition depends upon the geometry of the
Dirichlet fundamental domain representing the surface \DD/\Gamma
Stable Commutator Length in Amalgamated Free Products
We show that stable commutator length is rational on free products of free
Abelian groups amalgamated over , a class of groups containing
the fundamental groups of all torus knot complements. We consider a geometric
model for these groups and parameterize all surfaces with specified boundary
mapping to this space. Using this work we provide a topological algorithm to
compute stable commutator length in these groups. Further, we use the methods
developed to show that in free products of cyclic groups the stable commutator
length of a fixed varies quasirationally in the orders of the free factors.Comment: 28 pages, 5 figures. Corrected typographical errors, changed
exposition, added new results on quasirationalit
Double Bubbles Minimize
The classical isoperimetric inequality in R^3 states that the surface of
smallest area enclosing a given volume is a sphere. We show that the least area
surface enclosing two equal volumes is a double bubble, a surface made of two
pieces of round spheres separated by a flat disk, meeting along a single circle
at an angle of 120 degrees.Comment: 57 pages, 32 figures. Includes the complete code for a C++ program as
described in the article. You can obtain this code by viewing the source of
this articl
Inner geometry of complex surfaces: a valuative approach
Given a complex analytic germ in , the standard
Hermitian metric of induces a natural arc-length metric on , called the inner metric. We study the inner metric structure of the germ
of an isolated complex surface singularity by means of an infinite
family of numerical analytic invariants, called inner rates. Our main result is
a formula for the Laplacian of the inner rate function on a space of
valuations, the non-archimedean link of . We deduce in particular that
the global data consisting of the topology of , together with the
configuration of a generic hyperplane section and of the polar curve of a
generic plane projection of , completely determine all the inner rates
on , and hence the local metric structure of the germ. Several other
applications of our formula are discussed in the paper.Comment: Proposition 5.3 strengthened, exposition improved, some typos
corrected, references updated. 42 pages and 10 figures. To appear in Geometry
& Topolog
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