279 research outputs found
Dynamical Belyi maps
We study the dynamical properties of a large class of rational maps with
exactly three ramification points. By constructing families of such maps, we
obtain infinitely many conservative maps of degree ; this answers a question
of Silverman. Rather precise results on the reduction of these maps yield
strong information on the rational dynamics.Comment: 21 page
Optimal time travel in the Godel universe
Using the theory of optimal rocket trajectories in general relativity,
recently developed in arXiv:1105.5235, we present a candidate for the minimum
total integrated acceleration closed timelike curve in the Godel universe, and
give evidence for its minimality. The total integrated acceleration of this
curve is lower than Malament's conjectured value (Malament, 1984), as was
already implicit in the work of Manchak (Manchak, 2011); however, Malament's
conjecture does seem to hold for periodic closed timelike curves.Comment: 16 pages, 2 figures; v2: lower bound in the velocity and reference
adde
Doubly connected minimal surfaces and extremal harmonic mappings
The concept of a conformal deformation has two natural extensions:
quasiconformal and harmonic mappings. Both classes do not preserve the
conformal type of the domain, however they cannot change it in an arbitrary
way. Doubly connected domains are where one first observes nontrivial conformal
invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue
for quasiconformal and harmonic mappings, respectively. Combining these
concepts we obtain sharp estimates for quasiconformal harmonic mappings between
doubly connected domains. We then apply our results to the Cauchy problem for
minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a
sharp estimate of the modulus of a doubly connected minimal surface that
evolves from its inner boundary with a given initial slope.Comment: 35 pages, 2 figures. Minor edits, references adde
Contact numbers for congruent sphere packings in Euclidean 3-space
Continuing the investigations of Harborth (1974) and the author (2002) we
study the following two rather basic problems on sphere packings. Recall that
the contact graph of an arbitrary finite packing of unit balls (i.e., of an
arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is
the (simple) graph whose vertices correspond to the packing elements and whose
two vertices are connected by an edge if the corresponding two packing elements
touch each other. One of the most basic questions on contact graphs is to find
the maximum number of edges that a contact graph of a packing of n unit balls
can have in Euclidean 3-space. Our method for finding lower and upper estimates
for the largest contact numbers is a combination of analytic and combinatorial
ideas and it is also based on some recent results on sphere packings. Finally,
we are interested also in the following more special version of the above
problem. Namely, let us imagine that we are given a lattice unit sphere packing
with the center points forming the lattice L in Euclidean 3-space (and with
certain pairs of unit balls touching each other) and then let us generate
packings of n unit balls such that each and every center of the n unit balls is
chosen from L. Just as in the general case we are interested in finding good
estimates for the largest contact number of the packings of n unit balls
obtained in this way.Comment: 18 page
Topology and Signature Changes in Braneworlds
It has been believed that topology and signature change of the universe can
only happen accompanied by singularities, in classical, or instantons, in
quantum, gravity. In this note, we point out however that in the braneworld
context, such an event can be understood as a classical, smooth event. We
supply some explicit examples of such cases, starting from the
Dirac-Born-Infeld action. Topology change of the brane universe can be realised
by allowing self-intersecting branes. Signature change in a braneworld is made
possible in an everywhere Lorentzian bulk spacetime. In our examples, the
boundary of the signature change is a curvature singularity from the brane
point of view, but nevertheless that event can be described in a completely
smooth manner from the bulk point of view.Comment: 26 pages, 8 figures, references and comments are added, minor
revisions and a number of additional footnotes added, error corrected, minor
corrections, to appear in Class. Quant. Gra
Statistical mechanics approach to some problems in conformal geometry
A weak law of large numbers is established for a sequence of systems of N
classical point particles with logarithmic pair potential in \bbR^n, or
\bbS^n, n\in \bbN, which are distributed according to the configurational
microcanonical measure , or rather some regularization thereof,
where H is the configurational Hamiltonian and E the configurational energy.
When with non-extensive energy scaling E=N^2 \vareps, the
particle positions become i.i.d. according to a self-consistent Boltzmann
distribution, respectively a superposition of such distributions. The
self-consistency condition in n dimensions is some nonlinear elliptic PDE of
order n (pseudo-PDE if n is odd) with an exponential nonlinearity. When n=2,
this PDE is known in statistical mechanics as Poisson-Boltzmann equation, with
applications to point vortices, 2D Coulomb and magnetized plasmas and
gravitational systems. It is then also known in conformal differential
geometry, where it is the central equation in Nirenberg's problem of prescribed
Gaussian curvature. For constant Gauss curvature it becomes Liouville's
equation, which also appears in two-dimensional so-called quantum Liouville
gravity. The PDE for n=4 is Paneitz' equation, and while it is not known in
statistical mechanics, it originated from a study of the conformal invariance
of Maxwell's electromagnetism and has made its appearance in some recent model
of four-dimensional quantum gravity. In differential geometry, the Paneitz
equation and its higher order n generalizations have applications in the
conformal geometry of n-manifolds, but no physical applications yet for general
n. Interestingly, though, all the Paneitz equations have an interpretation in
terms of statistical mechanics.Comment: 17 pages. To appear in Physica
QCD Strings as Constrained Grassmannian Sigma Model:
We present calculations for the effective action of string world sheet in R3
and R4 utilizing its correspondence with the constrained Grassmannian sigma
model. Minimal surfaces describe the dynamics of open strings while harmonic
surfaces describe that of closed strings. The one-loop effective action for
these are calculated with instanton and anti-instanton background, reprsenting
N-string interactions at the tree level. The effective action is found to be
the partition function of a classical modified Coulomb gas in the confining
phase, with a dynamically generated mass gap.Comment: 22 pages, Preprint: SFU HEP-116-9
Surfaces immersed in su(N+1) Lie algebras obtained from the CP^N sigma models
We study some geometrical aspects of two dimensional orientable surfaces
arrising from the study of CP^N sigma models. To this aim we employ an
identification of R^(N(N+2)) with the Lie algebra su(N+1) by means of which we
construct a generalized Weierstrass formula for immersion of such surfaces. The
structural elements of the surface like its moving frame, the Gauss-Weingarten
and the Gauss-Codazzi-Ricci equations are expressed in terms of the solution of
the CP^N model defining it. Further, the first and second fundamental forms,
the Gaussian curvature, the mean curvature vector, the Willmore functional and
the topological charge of surfaces are expressed in terms of this solution. We
present detailed implementation of these results for surfaces immersed in su(2)
and su(3) Lie algebras.Comment: 32 pages, 1 figure; changes: major revision of presentation,
clarifications adde
A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space
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