403 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
Can one see the fundamental frequency of a drum?
We establish two-sided estimates for the fundamental frequency (the lowest
eigenvalue) of the Laplacian in an open subset G of R^n with the Dirichlet
boundary condition. This is done in terms of the interior capacitary radius of
G which is defined as the maximal possible radius of a ball B which has a
negligible intersection with the complement of G. Here negligibility of a
subset F in B means that the Wiener capacity of F does not exceed gamma times
the capacity of B, where gamma is an arbitrarily fixed constant between 0 and
1. We provide explicit values of constants in the two-sided estimates.Comment: 18 pages, some misprints correcte
Minimal surfaces bounded by elastic lines
In mathematics, the classical Plateau problem consists of finding the surface
of least area that spans a given rigid boundary curve. A physical realization
of the problem is obtained by dipping a stiff wire frame of some given shape in
soapy water and then removing it; the shape of the spanning soap film is a
solution to the Plateau problem. But what happens if a soap film spans a loop
of inextensible but flexible wire? We consider this simple query that couples
Plateau's problem to Euler's Elastica: a special class of twist-free curves of
given length that minimize their total squared curvature energy. The natural
marriage of two of the oldest geometrical problems linking physics and
mathematics leads to a quest for the shape of a minimal surface bounded by an
elastic line: the Euler-Plateau problem. We use a combination of simple
physical experiments with soap films that span soft filaments, scaling
concepts, exact and asymptotic analysis combined with numerical simulations to
explore some of the richness of the shapes that result. Our study raises
questions of intrinsic interest in geometry and its natural links to a range of
disciplines including materials science, polymer physics, architecture and even
art.Comment: 14 pages, 4 figures. Supplementary on-line material:
http://www.seas.harvard.edu/softmat/Euler-Plateau-problem
Aspects of Large N Gauge Theory Dynamics as Seen by String Theory
In this paper we explore some of the features of large N supersymmetric and
nonsupersymmetric gauge theories using Maldacena's duality conjectures. We
shall show that the resulting strong coupling behavior of the gauge theories is
consistent with our qualitative expectations of these theories. Some of these
consistency checks are highly nontrivial and give additional evidence for the
validity of the proposed dualities.Comment: 31 pages, LaTeX, 11 eps figures, typos correcte
Higher order Jordan Osserman Pseudo-Riemannian manifolds
We study the higher order Jacobi operator in pseudo-Riemannian geometry. We
exhibit a family of manifolds so that this operator has constant Jordan normal
form on the Grassmannian of subspaces of signature (r,s) for certain values of
(r,s). These pseudo-Riemannian manifolds are new and non-trivial examples of
higher order Osserman manifolds
On Uniqueness of Boundary Blow-up Solutions of a Class of Nonlinear Elliptic Equations
We study boundary blow-up solutions of semilinear elliptic equations
with , or with , where is a second order
elliptic operator with measurable coefficients. Several uniqueness theorems and
an existence theorem are obtained.Comment: To appear in Comm. Partial Differential Equations; 10 page
Curvature homogeneous spacelike Jordan Osserman pseudo-Riemannian manifolds
Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of
signature (2s,s) which are not locally homogeneous but whose curvature tensors
never the less exhibit a number of important symmetry properties. They are
curvature homogeneous; their curvature tensor is modeled on that of a local
symmetric space. They are spacelike Jordan Osserman with a Jacobi operator
which is nilpotent of order 3; they are not timelike Jordan Osserman. They are
k-spacelike higher order Jordan Osserman for ; they are k-timelike
higher order Jordan Osserman for , and they are not k timelike
higher order Jordan Osserman for .Comment: Update bibliography, fix minor misprint
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
Curvature properties of -null Osserman Lorentzian -manifolds
We expound some results about the relationships between the Jacobi operators
with respect to null vectors on a Lorentzian -manifold and the
Jacobi operators with respect to particular spacelike unit vectors on . We
study the number of the eigenvalues of such operators in a -null Osserman
Lorentzian -manifold, under suitable assumptions on the dimension
of the manifold. Then, we generalize a curvature characterization, previously
obtained by the first author for Lorentzian -null Osserman
-manifolds with exactly two characteristic vector fields, to the
case of those with an arbitrary number of characteristic vector fields.Comment: 15 pages; signs corrected on page 8, reference adde
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