12,565 research outputs found
Self-shrinkers with a rotational symmetry
In this paper we present a new family of non-compact properly embedded,
self-shrinking, asymptotically conical, positive mean curvature ends
that are hypersurfaces of revolution with
circular boundaries. These hypersurface families interpolate between the plane
and half-cylinder in , and any rotationally symmetric
self-shrinking non-compact end belongs to our family. The proofs involve the
global analysis of a cubic-derivative quasi-linear ODE. We also prove the
following classification result: a given complete, embedded, self-shrinking
hypersurface of revolution is either a hyperplane ,
the round cylinder of radius , the
round sphere of radius , or is diffeomorphic to an (i.e. a "doughnut" as in [Ang], which when is a torus). In
particular for self-shrinkers there is no direct analogue of the Delaunay
unduloid family. The proof of the classification uses translation and rotation
of pieces, replacing the method of moving planes in the absence of isometries.Comment: Trans. Amer. Math. Soc. (2011), to appear; 23 pages, 1 figur
Exceptional Laguerre polynomials
The aim of this paper is to present the construction of exceptional Laguerre
polynomials in a systematic way, and to provide new asymptotic results on the
location of the zeros. To describe the exceptional Laguerre polynomials we
associate them with two partitions. We find that the use of partitions is an
elegant way to express these polynomials and we restate some of their known
properties in terms of partitions. We discuss the asymptotic behavior of the
regular zeros and the exceptional zeros of exceptional Laguerre polynomials as
the degree tends to infinity.Comment: To appear in Studies in Applied Mathematic
Phase Space Reduction of Star Products on Cotangent Bundles
In this paper we construct star products on Marsden-Weinstein reduced spaces
in case both the original phase space and the reduced phase space are
(symplectomorphic to) cotangent bundles. Under the assumption that the original
cotangent bundle carries a symplectique structure of form
with a closed two-form on , is
equipped by the cotangent lift of a proper and free Lie group action on and
by an invariant star product that admits a -equivariant quantum momentum
map, we show that the reduced phase space inherits from a star product.
Moreover, we provide a concrete description of the resulting star product in
terms of the initial star product on and prove that our reduction scheme
is independent of the characteristic class of the initial star product. Unlike
other existing reduction schemes we are thus able to reduce not only strongly
invariant star products. Furthermore in this article, we establish a relation
between the characteristic class of the original star product and the
characteristic class of the reduced star product and provide a classification
up to -equivalence of those star products on , which
are invariant with respect to a lifted Lie group action. Finally, we
investigate the question under which circumstances `quantization commutes with
reduction' and show that in our examples non-trivial restrictions arise
The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces.
Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c0 and ℓp for 1p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E(ℓ2n)c0, that is, E is the c0-direct sum of the finite-dimensional Hilbert spaces ℓ21,ℓ22,…,ℓ2n,…
Mean curvature self-shrinkers of high genus: Non-compact examples
We give the first rigorous construction of complete, embedded self-shrinking
hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The
surfaces exist for any sufficiently large prescribed genus , and are
non-compact with one end. Each has symmetries and comes from
desingularizing the intersection of the plane and sphere through a great
circle, a configuration with very high symmetry. Each is at infinity asymptotic
to the cone in over a -periodic graph on an equator
of the unit sphere , with the shape of a
periodically "wobbling sheet". This is a dramatic instability phenomenon, with
changes of asymptotics that break much more symmetry than seen in minimal
surface constructions. The core of the proof is a detailed understanding of the
linearized problem in a setting with severely unbounded geometry, leading to
special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the
gradient terms. This involves identifying new, adequate weighted H\"older
spaces of asymptotically conical functions in which the operators invert, via a
Liouville-type result with precise asymptotics.Comment: 41 pages, 1 figure; minor typos fixed; to appear in J. Reine Angew.
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