We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi–Yau manifolds. For example, we prove that given any real-analytic one parameter family of Riemannian metrics g t on a three-dimensional manifold Y with volume form independent of t and with a real-analytic family of nowhere vanishing harmonic one forms θ t , then (Y,g t ) can be realized as a family of special Lagrangian submanifolds of a Calabi–Yau manifold X. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus action. We use this to construct examples of n-parameter families of special Lagrangian tori inside n + k-dimensional Calabi–Yau manifolds with torus symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian fibrations with T 2-symmetry.
Isometric Embeddings of Families of Special Lagrangian Submanifolds / D. Matessi. - In: ANNALS OF GLOBAL ANALYSIS AND GEOMETRY. - ISSN 0232-704X. - 29:3(2006), pp. 197-220. [10.1007/s10455-005-9008-2]
Isometric Embeddings of Families of Special Lagrangian Submanifolds
D. MatessiPrimo
2006
Abstract
We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi–Yau manifolds. For example, we prove that given any real-analytic one parameter family of Riemannian metrics g t on a three-dimensional manifold Y with volume form independent of t and with a real-analytic family of nowhere vanishing harmonic one forms θ t , then (Y,g t ) can be realized as a family of special Lagrangian submanifolds of a Calabi–Yau manifold X. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus action. We use this to construct examples of n-parameter families of special Lagrangian tori inside n + k-dimensional Calabi–Yau manifolds with torus symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian fibrations with T 2-symmetry.File | Dimensione | Formato | |
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