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A characterization of quasiconformal maps in terms of sets of finite perimeter

Jones, Rebekah
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1560867563841096

Abstract Details

2019, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
In Euclidean space, it is well-known that quasiconformal maps are characterized by the quasi-preservation of the n-modulus of curves. This fact is also known in the setting of Ahlfors Q-regular metric measure spaces. In this dissertation, we present a similar characterization of quasiconformal maps on metric measure spaces in terms of the modulus of collections of surfaces. We make some geometric assumptions on the metric measure spaces, namely that they are complete, Ahlfors Q-regular and support a 1-Poincare inequality. We show that quasiconformal maps in this setting quasi-preserve the Q/(Q-1)-modulus of the (Q-1)-Hausdorff measures restricted to the S boundary of E. Here the S boundary of E is the collection of points for which E and E complement have positive lower density, which is a subset of the measure theoretic boundary. When E is a set of finite perimeter, the (Q-1)-Hausdorff measure restricted to SE is comparable to the perimeter measure of E. This result was shown in Euclidean spaces for a more restrictive class of surfaces by Kelly. There, the sets of finite perimeter are taken to be sets whose reduced boundary has finite (n-1)-Hausdorff measure, which is a non-standard definition. Conversely, we show that a homeomorphism that quasi-preserves the modulus of sets of finite perimeter is necessarily quasiconformal. To do so, we show that there is a duality between the Q-modulus of curves and the Q/(Q-1)-modulus of surfaces, generalizing a result in Euclidean space of Aikawa and Ohtsuka.
Nageswari Shanmugalingam, Ph.D. (Committee Chair)
Panu Kalevi Lahti, Ph.D. (Committee Member)
Andrew Lorent, Ph.D. (Committee Member)
Gareth Speight, Ph.D. (Committee Member)
90 p.
Mathematics
quasiconformal; finite perimeter; modulus of curves; modulus of surfaces; perimeter measure; metric measure space

Recommended Citations

Citations

  • Jones, R. (2019). A characterization of quasiconformal maps in terms of sets of finite perimeter [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1560867563841096

    APA Style (7th edition)

  • Jones, Rebekah. A characterization of quasiconformal maps in terms of sets of finite perimeter. 2019. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1560867563841096.

    MLA Style (8th edition)

  • Jones, Rebekah. "A characterization of quasiconformal maps in terms of sets of finite perimeter." Doctoral dissertation, University of Cincinnati, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1560867563841096

    Chicago Manual of Style (17th edition)

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This open access ETD is published by University of Cincinnati and OhioLINK.