645 research outputs found
Wrinkled fibrations on near-symplectic manifolds
Motivated by the programmes initiated by Taubes and Perutz, we study the
geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed
2-form which is symplectic outside a union of embedded 1-dimensional
submanifolds, and broken Lefschetz fibrations on them. We present a set of four
moves which allow us to pass from any given fibration to any other broken
fibration which is deformation equivalent to it. Moreover, we study the change
of the near-symplectic geometry under each of these moves. The arguments rely
on the introduction of a more general class of maps, which we call wrinkled
fibrations and which allow us to rely on classical singularity theory.Finally,
we illustrate these constructions by showing how one can merge components of
the zero-set of the near-symplectic form. We also disprove a conjecture of Gay
and Kirby by showing that any achiral broken Lefschetz fibration can be turned
into a broken Lefschetz fibration by applying a sequence of our moves.Comment: 35 pages, 12 figures. Final version. Minor corrections and
clarification
Hyperbolic Unfoldings of Minimal Hypersurfaces
We study the intrinsic geometry of area minimizing (and also of almost
minimizing) hypersurfaces from a new point of view by relating this subject to
quasiconformal geometry. For any such hypersurface we define and construct a
so-called S-structure which reveals some unexpected geometric and analytic
properties of the hypersurface and its singularity set. In this paper, this is
used to prove the existence of hyperbolic unfoldings: canonical conformal
deformations of the regular part of these hypersurfaces into complete Gromov
hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to the
singular set
Versal unfoldings for linear retarded functional differential equations
We consider parametrized families of linear retarded functional differential
equations (RFDEs) projected onto finite-dimensional invariant manifolds, and
address the question of versality of the resulting parametrized family of
linear ordinary differential equations. A sufficient criterion for versality is
given in terms of readily computable quantities. In the case where the
unfolding is not versal, we show how to construct a perturbation of the
original linear RFDE (in terms of delay differential operators) whose
finite-dimensional projection generates a versal unfolding. We illustrate the
theory with several examples, and comment on the applicability of these results
to bifurcation analyses of nonlinear RFDEs
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