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Deformations of special Lagrangian submanifolds, DPhil thesis
, 2002
"... In this thesis we study the deformations of special Lagrangian submanifolds X ⊆ M sitting inside a CalabiYau manifold (M, g, J,Ω). Let N be the normal bundle of X, and identify N ∼ = T ∗X via the complex structure J and induced metric on X. Then using the exponential map one can identify small 1fo ..."
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In this thesis we study the deformations of special Lagrangian submanifolds X ⊆ M sitting inside a CalabiYau manifold (M, g, J,Ω). Let N be the normal bundle of X, and identify N ∼ = T ∗X via the complex structure J and induced metric on X. Then using the exponential map one can identify small 1forms ξ on X with submanifolds Xξ ⊆M close to X. In the case that X is compact, McLean [50, Theorem 36], showed that the small 1forms ξ parameterising special Lagrangian submanifolds Xξ ⊆ M form a smooth manifold M ⊆ C∞(T ∗X) of dimension b1(X), the first Betti number of X. We give a full proof of this result, including the necessary details which were absent from [50]. In fact our result Theorem 3.21 is an extension of the original McLean theorem, in that we show that the special Lagrangian deformationsM persist under (certain types of) perturbations of the ambient CalabiYau structure. We then go on to consider the situation when X ⊆ Cn is noncompact, but asymptotic to a cone C ⊆ Cn at a specified rate α ̃ < 1 of decay. Provided that α ̃ is not too negative, it turns out that for almost all α ̃ there is again a smooth manifold Mα ̃ ⊆ C∞(T ∗X) parameterising the special Lagrangian submanifolds Xξ ⊆ Cn which are near to X and decay towards C at rate α̃. The main result here is Theorem 6.45, which also gives the dimensions of the smooth manifold Mα̃. It turns out that for small rates of decay, dimMα ̃ depends only on the topology of X, whereas for higher rates dimMα ̃ will also depend on analytic data got from the link Σ: = S2n−1 ∩ C of the cone C. Along the way to proving Theorem 6.45 we develop a theory of analysis for asymptotically conical Riemannian manifolds, expanding on the existing theory of Lockhart and McOwen [46] and Lockhart [45] for damped Sobolev spaces. In particular, in Section 6.1.1 we give the relevant details for damped Hölder spaces. We finish in Section 6.3 by applying our theory to some specific examples, and prove the existence of special Lagrangian submanifolds in Xξ ⊆ Cn which were previously unknown. i
CONSTRUCTION OF HAMILTONIANMINIMAL LAGRANGIAN SUBMANIFOLDS IN COMPLEX EUCLIDEAN SPACE
, 2009
"... We describe several families of Lagrangian submanifolds in the complex Euclidean space which are Hminimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendri ..."
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We describe several families of Lagrangian submanifolds in the complex Euclidean space which are Hminimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of the odddimensional unit sphere.
On indefinite special Lagrangian submanifolds in indefinite complex Euclidean spaces
 Journal of Geometry and Physics
"... Abstract. In this paper, we show that the calibrated method can also be used to detect indefinite minimal Lagrangian submanifolds in Cm k. We introduce the notion of indefinite special Lagrangian submanifolds in Cm k and generalize the wellknown work of HarveyLawson to the indefinite case. 1. ..."
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Abstract. In this paper, we show that the calibrated method can also be used to detect indefinite minimal Lagrangian submanifolds in Cm k. We introduce the notion of indefinite special Lagrangian submanifolds in Cm k and generalize the wellknown work of HarveyLawson to the indefinite case. 1.
Minimal Lagrangian submanifolds in the complex hyperbolic space
, 2001
"... In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1parameter families with cohomogeneity one. We characterize them as the only minimal Lagrangian submanifolds in CH n foliated by umbilical hypersurf ..."
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Cited by 5 (2 self)
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In this paper we construct new examples of minimal Lagrangian submanifolds in the complex hyperbolic space with large symmetry groups, obtaining three 1parameter families with cohomogeneity one. We characterize them as the only minimal Lagrangian submanifolds in CH n foliated by umbilical hypersurfaces of Lagrangian subspaces RH n of CH n. Several suitable generalizations of the above construction allow us to get new families of minimal Lagrangian submanifolds in CH n from curves in CH 1 and (n − 1)dimensional minimal Lagrangian submanifolds of the complex space forms CP n−1, CH n−1 and C n−1. Similar constructions are made in the complex projective space CP n. 1
Intersections of quadrics, momentangle manifolds, and Hamiltonianminimal Lagrangian embeddings
, 2011
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Finitegap minimal Lagrangian surfaces
 in CP 2 , in Riemann Surfaces, Harmonic Maps and Visualization, OCAMI Stud
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Closed twisted products and SO(p)× SO(q) invariant special Lagrangian cones
 Comm. Anal. Geom
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