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VARIATIONAL PROBLEMS OF NORMAL CURVATURE TENSOR AND CONCIRCULAR SCALAR FIELDS

  • SAKAMOTO KUNIO
    Department of Mathematics, Faculty of Science, Saitama University

Abstract

We consider the integral of (the square of) the length of the normal curvature tensor for immersions of manifolds into real space forms, especially into spheres. The first variation formula is given and the Euler-Lagrange equation is expressed in terms of the isothermal coordinates when the submanifold is two-dimensional. The relations between the critical surfaces and Willmore surfaces are discussed. We also give formulas concerning the residue of logarithmic singularities of $S$-Willmore points or estimate it by a conformal invariant.<br>We show that if a compact critical surface satisfies certain conditions and the immersion is minimal, then the Gauss curvature is a non-negative constant and the immersion is a standard minimal immersion of a sphere or a constant isotropic minimal immersion of a flat torus. To prove this result, we study two-dimensional Riemannian manifolds admitting concircular scalar fields whose characteristic functions are polynomials of degree $2$. Moreover, the case that the characteristic functions are polynomials of degree $3$ is studied.

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Details

  • CRID
    1390282680093323264
  • NII Article ID
    110000027009
  • DOI
    10.2748/tmj/1113246939
  • ISSN
    2186-585X
    0040-8735
  • MRID
    1979497
  • Text Lang
    en
  • Data Source
    • JaLC
    • CiNii Articles
    • Crossref

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