Nonexistence of minimal Lagrangian spheres in hyperKähler manifolds

Smoczyk, Knut

doi:10.1007/pl00013454

Cited by 3 publications

(4 citation statements)

References 11 publications

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“…We point out that part (2) in Theorem 22 is an improvement of the result by Smoczyk obtained in [50]. He proved Theorem 22 for s = 1 without using spin geometry.…”

Section: The Hodge Laplacian On Minimal Lagrangian Submanifoldsmentioning

confidence: 65%

“…In this situation, the restriction to the submanifold of the spin bundle on the Calabi-Yau manifold is nothing but the complex exterior bundle, the restriction of the spin connection induces the Levi-Cività connection and the induced Dirac operator is just the Euler operator on the submanifold (Proposition 20). Using these identifications, we see that the restriction to the submanifold of parallel spinor fields of the Calabi-Yau manifold are harmonic forms, which yield sharp topological restrictions on such submanifolds (Theorem 22), improving those obtained in [50].…”

Section: Introductionmentioning

confidence: 74%

“…On the other hand, Smoczyk [50] showed that non-trivial harmonic 2-forms on minimal Lagrangian submanifolds of hyper-Kähler manifolds are obtained as restrictions to the submanifold of some parallel exterior forms on the ambient space. These two works can be considered as the starting point of this paper which can be framed into a series of results where well-behaved geometric objects on submanifolds are obtained from parallel objects on the ambient manifold.…”

Section: Introductionmentioning

confidence: 99%

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Spinc geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

2006

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Abstract. From the existence of parallel spinor fields on CalabiYau, hyper-Kähler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers. When the ambient manifold is Kähler-Einstein with positive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kählerian Killing spinor fields for some particular spin c structures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. Applications on the Morse index of minimal Lagrangian submanifolds are obtained.

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“…We point out that part (2) in Theorem 22 is an improvement of the result by Smoczyk obtained in [50]. He proved Theorem 22 for s = 1 without using spin geometry.…”

Section: The Hodge Laplacian On Minimal Lagrangian Submanifoldsmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

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Spinc geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

2006

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“…Since minimal Lagrangian cones are relevant in the study of singularities for minimal Lagrangian submanifolds it is clear that minimal Legendrian submanifolds of S 2n+1 are important as well. In [5] we proved that the restriction of a parallel, anti-compatible k-form on a Kähler manifold to a minimal Lagrangian immersion becomes a harmonic k-form. In particular this result implied the nonexistence of minimal, orientable, closed Lagrangian immersions L in hyperKähler manifolds if the second Betti number of L equals 0.…”

mentioning

confidence: 98%

Note on the spectrum of the Hodge-Laplacian for k -forms on minimal Legendre submanifolds in $S^{2n+1}$

Smoczyk

2002

Calculus of Variations and Partial Differential Equations

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Given a minimal Legendre immersion L in S 2n+1 and n ≥ k ≥ 1 we prove that n + 1 − k is an eigenvalue of the Hodge-Laplacian acting on k and (k − 1)-forms on L.In particular we show that the eigenspaces Eig k (n + 1 − k) and Eig k−1 (n + 1 − k) are at least of dimension n k PreliminariesA contact manifold (of restricted type) 1 (M, λ) is an odd-dimensional manifold of dimension 2n + 1 together with a one-form λ such that λ ∧ (dλ) n defines a volume form on M . One observes that a contact manifold is orientable and that the contact form λ defines a natural orientation.Assume now that (M, λ) is a given contact manifold of dimension 2n + 1. λ defines a 2n-dimensional vector bundle ξ over M , where at each point p ∈ M the fiber ξ p of ξ is given byMoreover, since λ ∧ (dλ) n is a volume form, we see that ω := dλ |ξ⊕ξ is a closed non degenerate two-form and hence defines a symplectic product on ξ so that (ξ, ω) becomes a symplectic vector bundle. Since the dimension of M is odd, the two-form dλ must be degenerate on T M. Therefore one obtains a line bundle l over M via the definitionMore generally a contact manifold M is a differentiable manifold of odd dimension 2n + 1 with a completely nonintegrable distribution ξ of hyperplanes in the tangent space. Locally such hyperplane fields can be described as the kernel of a nonvanishing one-form λ. The nonintegrability then implies that λ ∧ (dλ) n locally defines a volume form. If this one-form λ exists globally then we speak of a contact manifold of restricted type.

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Mean curvature flow of area decreasing maps between Riemann surfaces

Savas-Halilaj

Smoczyk

2017

Ann Glob Anal Geom

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Nonexistence of minimal Lagrangian spheres in hyperKähler manifolds

Cited by 3 publications

References 11 publications

Spinc geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

Spinc geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds

Note on the spectrum of the Hodge-Laplacian for k -forms on minimal Legendre submanifolds in $S^{2n+1}$

Mean curvature flow of area decreasing maps between Riemann surfaces

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