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6 results

  • 8 - Constrained Willmore Surfaces and the Isothermic Surface Condition

  • Áurea Casinhas Quintino, Universidade Nova de Lisboa, Portugal
  • Book:
    Constrained Willmore Surfaces
    Published online:
    13 May 2021
    Print publication:
    10 June 2021, pp 145-182

  • Summary

    This chapter is dedicated to the very special class of constant mean curvature surfaces. A classical result by Thomsen characterizes isothermic Willmore surfaces in 3-space as minimal surfaces in some 3-dimensional space-form. Constant mean curvature surfaces in 3-dimensional space-forms are examples of constrained Willmore surfaces, characterized by the existence of some conserved quantity. Both constrained Willmore spectral deformation and Bäcklund transformation prove to preserve the existence of such a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, in the latter case. The class of constant mean curvature surfaces in 3-dimensional space-forms lies, in this way, at the intersection of several integrable geometries, with classical transformations of its own, as well as transformations as a class of constrained Willmore surfaces, together with transformations as a subclass of the class of isothermic surfaces, as we explore in this chapter. Constrained Willmore transformation proves to be unifying to this rich transformation theory, as we shall conclude.

  • 9 - The Special Case of Surfaces in 4-Space

  • Áurea Casinhas Quintino, Universidade Nova de Lisboa, Portugal
  • Book:
    Constrained Willmore Surfaces
    Published online:
    13 May 2021
    Print publication:
    10 June 2021, pp 183-234

  • Summary

    This chapter is dedicated to the special case of surfaces in 4-space. Our approach is quaternionic, based on the model of the conformal 4-sphere on the quaternionic projective space. We extend the Darboux transformation of Willmore surfaces in 4-space presented by Burstall–Ferus–Leschke–Pedit–Pinkall, based on the solution of a Riccati equation, to a transformation of constrained Willmore surfaces in 4-space into new ones. We prove that non-trivial Darboux transformation of constrained Willmore surfaces in 4-space can be obtained as a particular case of Bäcklund transformation. This Darboux transformation of constrained Willmore surfaces displays a striking similarity with the description of isothermic Darboux transformation of constant mean curvature surfaces in Euclidean 3-space presented by Hertrich-Jeromin−Pedit, which, in fact, proves to be obtainable as a particular case of constrained Willmore Bäcklund transformation.

Constrained Willmore Surfaces
  • Symmetries of a Möbius Invariant Integrable System
  • Áurea Casinhas Quintino
  • Published online:
    13 May 2021
    Print publication:
    10 June 2021
  • From Bäcklund to Darboux, this monograph presents a comprehensive journey through the transformation theory of constrained Willmore surfaces, a topic of great importance in modern differential geometry and, in particular, in the field of integrable systems in Riemannian geometry. The first book on this topic, it discusses in detail a spectral deformation, Bäcklund transformations and Darboux transformations, and proves that all these transformations preserve the existence of a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, and bridging the gap between different approaches to the subject, classical and modern. Clearly written with extensive references, chapter introductions and self-contained accounts of the core topics, it is suitable for newcomers to the theory of constrained Wilmore surfaces. Many detailed computations and new results unavailable elsewhere in the literature make it also an appealing reference for experts.

  • Stochastic LU factorizations, Darboux transformations and urn models

  • Part of
  • F. Alberto Grünbaum, Manuel D. de la Iglesia
  • Journal:
    Journal of Applied Probability / Volume 55 / Issue 3 / September 2018
    Published online by Cambridge University Press:
    16 November 2018, pp. 862-886
    Print publication:
    September 2018

  • We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

  • TRANSFORMATION OF STURM–LIOUVILLE PROBLEMS WITH DECREASING AFFINE BOUNDARY CONDITIONS

  • Paul A. Binding, Patrick J. Browne, Warren J. Code, Bruce A. Watson
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 47 / Issue 3 / October 2004
    Published online by Cambridge University Press:
    09 November 2004, pp. 533-552
    • Article
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  • We consider Sturm–Liouville boundary-value problems on the interval $[0,1]$ of the form $-y''+qy=\lambda y$ with boundary conditions $y'(0)\sin\alpha=y(0)\cos\alpha$ and $y'(1)=(a\lambda+b)y(1)$, where $a\lt0$. We show that via multiple Crum–Darboux transformations, this boundary-value problem can be transformed ‘almost’ isospectrally to a boundary-value problem of the same form, but with the boundary condition at $x=1$ replaced by $y'(1)\sin\beta=y(1)\cos\beta$, for some $\beta$.

    AMS 2000 Mathematics subject classification: Primary 34B07; 47E05; 34L05

  • Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting

  • G. F. Helminck, J. W. van de Leur
  • Journal:
    Canadian Journal of Mathematics / Volume 53 / Issue 2 / 01 April 2001
    Published online by Cambridge University Press:
    20 November 2018, pp. 278-309
    Print publication:
    01 April 2001
    • Article
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  • In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the $\text{KP}$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of $\tau $-functions is given.