Browsing by Author "Gevirtz, J"

Now showing 1 - 4 of 4
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    Constant principal strain mappings on 2-manifolds
    (2000) Chuaqui, M; Gevirtz, J
    We study mappings between Riemannian 2-manifolds which have constant principal stretching factors (cps-mappings). Such mappings f can be described in terms of the relationship between the geodesic curvature of the curves of principal strain at p and that of their images at f (p). In the context of local coordinates this relationship takes the form of a nonlinear hyperbolic system, the blow-up properties of which depend on the Gaussian curvatures of the two manifolds. We use the theory of such systems to study global existence when both manifolds are the hyperbolic plane H-2 and obtain a simple description of all cps-mappings of H-2 onto itself. We also obtain a distortion result for disks in H-2 as well as some nonexistence results for cps-mappings of the Euclidean plane onto certain classes of manifolds. In addition, our treatment of cps-mappings in H-2 yields, virtually as a corollary, a generalization of a theorem of Epstein to the effect that a curve in hyperbolic n-space whose geodesic curvature is bounded by 1 must be simple.
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    Hencky-Prandtl nets and constant principal strain mappings with isolated singularities
    (2000) Gevirtz, J
    The work presented in this paper is motivated in large measure by the appearance of Hencky-Prandtl nets (HP-nets) in the context of planar quasi-isometries With constant principal stretching factors (cps-mappings) and by compelling analogies between such mappings and those given by analytic functions of one complex variable. We study the behavior of HP-nets in the vicinity of isolated singularities and use the results of this analysis to show that if an HP-net is regular in the entire plane except for isolated singularities, then it can have at most two of them, and that all possible nets of this kind fall into five classes each of which depends on a small number of parameters. In light of the relationship between HP-nets and cps-mappings it follows that an analogous statement holds for the latter as well, and this connection is further exploited to prove that HP-nets regular except for isolated singularities in smoothly bounded Jordan domains have nontangential limits in the appropriate sense at almost all boundary points. The treatment includes, in addition, an interpretation of cps-mappings with isolated singularities as deformations produced by the cryptocrystalline solidification:with microscopic flaws of a planar film and a discussion of the problem of just how the singularities of such mappings can actually be distributed in a given domain.
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    Simple curves in Rn and Ahlfors' Schwarzian derivative
    (2004) Chuaqui, M; Gevirtz, J
    We derive sharp injectivity criteria for mappings f : ( 1; 1) --> R-n in terms of Ahlfors' definition of the Schwarzian derivative for such mappings.
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    Singularity sets of constant principal strain deformations
    (2001) Gevirtz, J
    We show that if f is a mapping with constant principal strains (cps-mapping) of a planar domain of the form D\S, where D is itself a domain and S is a closed subset of D with linear measure 0, then f has an extension to a cps-mapping of D\S', where S' subset of S has no accumulation points in D. The proof uses properties of cps-mappings attributable to the nonlinear hyperbolic nature of the underlying system of partial differential equations as well as results about their behavior in neighborhoods of isolated singularities previously established by the author. (C) 2001 Academic Press.