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  • PINCHING THEOREMS FOR A COMPACT MINIMAL SUBMANIFOLD IN A COMPLEX PROJECTIVE SPACE

  • Part of
  • MAYUKO KON
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 77 / Issue 1 / February 2008
    Published online by Cambridge University Press:
    01 February 2008, pp. 99-114
    Print publication:
    February 2008
    • Article
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  • We give a formula for the Laplacian of the second fundamental form of an n-dimensional compact minimal submanifold M in a complex projective space CPm. As an application of this formula, we prove that M is a geodesic minimal hypersphere in CPm if the sectional curvature satisfies K≥1/n, if the normal connection is flat, and if M satisfies an additional condition which is automatically satisfied when M is a CR submanifold. We also prove that M is the complex projective space CPn/2 if K≥3/n, and if the normal connection of M is semi-flat.