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2 results

  • 5 - The Euler–Lagrange ConstrainedWillmore Surface Equation

  • Áurea Casinhas Quintino, Universidade Nova de Lisboa, Portugal
  • Book:
    Constrained Willmore Surfaces
    Published online:
    13 May 2021
    Print publication:
    10 June 2021, pp 73-86

  • Summary

    In this chapter, we introduce constrained Willmore surfaces, the generalization of Willmore surfaces that arises when we consider critical points of the Willmore functional with respect to infinitesimally conformal variations, rather than with respect to all variations. Constrained Willmore surfaces in space-forms constitute a Möbius invariant class of surfaces with strong links to the theory of integrable systems, which we shall explore throughout this work. The introduction of a constraint in the variational problem equips surfaces with Lagrange multipliers, as first proven by Burstall–Pedit–Pinkall, in a manifestly conformally invariant characterization of constrained Willmore surfaces in space-forms in terms of the Hopf differential and the Schwarzian derivative, which we deduce in this chapter. The class of constrained Willmore surfaces was later given yet another manifestly conformally invariant characterization, free of holomorphic charts, by Burstall–Calderbank, which we derive in this chapter from the variational problem.

  • OSCILLATION OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

  • Part of
  • MARTIN CHUAQUI, PETER DUREN, BRAD OSGOOD, DENNIS STOWE
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 79 / Issue 1 / February 2009
    Published online by Cambridge University Press:
    09 February 2009, pp. 161-169
    Print publication:
    February 2009
    • Article
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  • In this note we study the zeros of solutions of differential equations of the form u′′+pu=0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given.