59 research outputs found

    Thurston's pullback map on the augmented Teichm\"uller space and applications

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    Let ff be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map σf\sigma_f of a finite-dimensional Teichm\"uller space. We prove that this map extends continuously to the augmented Teichm\"uller space and give an explicit construction for this extension. This allows us to characterize the dynamics of Thurston's pullback map near invariant strata of the boundary of the augmented Teichm\"uller space. The resulting classification of invariant boundary strata is used to prove a conjecture by Pilgrim and to infer further properties of Thurston's pullback map. Our approach also yields new proofs of Thurston's theorem and Pilgrim's Canonical Obstruction theorem.Comment: revised version, 28 page

    Minimal immersions of closed surfaces in hyperbolic three-manifolds

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    We study minimal immersions of closed surfaces (of genus g2g \ge 2) in hyperbolic 3-manifolds, with prescribed data (σ,tα)(\sigma, t\alpha), where σ\sigma is a conformal structure on a topological surface SS, and αdz2\alpha dz^2 is a holomorphic quadratic differential on the surface (S,σ)(S,\sigma). We show that, for each t(0,τ0)t \in (0,\tau_0) for some τ0>0\tau_0 > 0, depending only on (σ,α)(\sigma, \alpha), there are at least two minimal immersions of closed surface of prescribed second fundamental form Re(tα)Re(t\alpha) in the conformal structure σ\sigma. Moreover, for tt sufficiently large, there exists no such minimal immersion. Asymptotically, as t0t \to 0, the principal curvatures of one minimal immersion tend to zero, while the intrinsic curvatures of the other blow up in magnitude.Comment: 16 page

    Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces

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    Using Polyakov's functional integral approach with the Liouville action functional defined in \cite{ZT2} and \cite{LTT}, we formulate quantum Liouville theory on a compact Riemann surface X of genus g > 1. For the partition function and for the correlation functions with the stress-energy tensor components , we describe Feynman rules in the background field formalism by expanding corresponding functional integrals around a classical solution - the hyperbolic metric on X. Extending analysis in \cite{LT1,LT2,LT-Varenna,LT3}, we define the regularization scheme for any choice of global coordinate on X, and for Schottky and quasi-Fuchsian global coordinates we rigorously prove that one- and two-point correlation functions satisfy conformal Ward identities in all orders of the perturbation theory. Obtained results are interpreted in terms of complex geometry of the projective line bundle \cE_{c}=\lambda_{H}^{c/2} over the moduli space Mg\mathfrak{M}_{g}, where c is the central charge and λH\lambda_{H} is the Hodge line bundle, and provide Friedan-Shenker \cite{FS} complex geometry approach to CFT with the first non-trivial example besides rational models.Comment: 67 pages, 4 figures (Typos corrected as in the published version

    On the appearance of Eisenstein series through degeneration

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    Let Γ\Gamma be a Fuchsian group of the first kind acting on the hyperbolic upper half plane H\mathbb H, and let M=Γ\HM = \Gamma \backslash \mathbb H be the associated finite volume hyperbolic Riemann surface. If γ\gamma is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If γ\gamma is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If γΓ\gamma \in \Gamma corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.Comment: 15 pages, 2 figures. This paper has been accepted for publication in Commentarii Mathematici Helvetic

    Growth Based Morphogenesis of Vertebrate Limb Bud

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    Many genes and their regulatory relationships are involved in developmental phenomena. However, by chemical information alone, we cannot fully understand changing organ morphologies through tissue growth because deformation and growth of the organ are essentially mechanical processes. Here, we develop a mathematical model to describe the change of organ morphologies through cell proliferation. Our basic idea is that the proper specification of localized volume source (e.g., cell proliferation) is able to guide organ morphogenesis, and that the specification is given by chemical gradients. We call this idea “growth-based morphogenesis.” We find that this morphogenetic mechanism works if the tissue is elastic for small deformation and plastic for large deformation. To illustrate our concept, we study the development of vertebrate limb buds, in which a limb bud protrudes from a flat lateral plate and extends distally in a self-organized manner. We show how the proportion of limb bud shape depends on different parameters and also show the conditions needed for normal morphogenesis, which can explain abnormal morphology of some mutants. We believe that the ideas shown in the present paper are useful for the morphogenesis of other organs

    Fitness distributions and GA hardness

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    Considerable research effort has been spent in trying to formulate a good definition of GA-Hardness. Given an instance of a problem, the objective is to estimate the performance of a GA. Despite partial successes current definitions are still unsatisfactory. In this paper we make some steps towards a new, more powerful way of assessing problem difficulty based on the properties of a problem's fitness distribution. We present experimental results that strongly support this idea
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