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Nonexistence of minimal Lagrangian spheres in hyperKähler manifolds


We prove that for \(n>1\) one cannot immerse \(S^{2n}\) as a minimal Lagrangian manifold into a hyperKähler manifold. More generally we show that any minimal Lagrangian immersion of an orientable closed manifold \(L^{2n}\) into a hyperKähler manifold \(H^{4n}\) must have nonvanishing second Betti number \(\beta_2\) and that if \(\beta_2=1\), \(L^{2n}\) is a Kähler manifold and more precisely a Kähler submanifold in \(H^{4n}\) w.r.t. one of the complex structures on \(H^{4n}\). In addition we derive a result for the other Betti numbers.

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Received February 10, 1999 / Accepted April 23, 1999

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Smoczyk, K. Nonexistence of minimal Lagrangian spheres in hyperKähler manifolds. Calc Var 10, 41–48 (2000).

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  • Betti Number
  • Lagrangian Manifold
  • Lagrangian Immersion
  • Lagrangian Sphere
  • Minimal Lagrangian Immersion