Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings

Mironov, Andrey E.; Panov, Taras

doi:10.1007/s10688-013-0005-0

Cited by 23 publications

(15 citation statements)

References 26 publications

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“…These systems of quadrics are the same as those we used to define moment-angle manifolds, and therefore one can apply toric methods for analysing the topological structure of N . In [47] an effective criterion was obtained for N C m to be an embedding: the polytope corresponding to the intersection of quadrics must be Delzant. As a consequence, any Delzant polytope gives rise to an H-minimal Lagrangian submanifold N ⊂ C m .…”

Section: Hamiltonian-minimal Lagrangian Submanifoldsmentioning

confidence: 99%

Geometric structures on moment-angle manifolds

Panov¹

2013

Russ. Math. Surv.

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Abstract. The moment-angle complex Z K is cell complex with a torus action constructed from a finite simplicial complex K. When this construction is applied to a triangulated sphere K or, in particular, to the boundary of a simplicial polytope, the result is a manifold. Moment-angle manifolds and complexes are central objects in toric topology, and currently are gaining much interest in homotopy theory, complex and symplectic geometry.The geometric aspects of the theory of moment-angle complexes are the main theme of this survey. We review constructions of non-Kähler complexanalytic structures on moment-angle manifolds corresponding to polytopes and complete simplicial fans, and describe invariants of these structures, such as the Hodge numbers and Dolbeault cohomology rings. Symplectic and Lagrangian aspects of the theory are also of considerable interest. Moment-angle manifolds appear as level sets for quadratic Hamiltonians of torus actions, and can be used to construct new families of Hamiltonian-minimal Lagrangian submanifolds in a complex space, complex projective space or toric varieties.

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Section: Hamiltonian-minimal Lagrangian Submanifoldsmentioning

confidence: 99%

Geometric structures on moment-angle manifolds

Panov¹

2013

Russ. Math. Surv.

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“…Theorem 2 [3]. The immersion i : N C m is an embedding if and only if the associated P is a Delzant polyhedron.…”

Section: Hamiltonian-minimal Lagrangian Submanifolds In Toric Varietiesmentioning

confidence: 98%

“…In [2] and [3] the authors defined and studied a family of H-minimal Lagrangian submanifolds of C m arising from intersections of real quadrics. Here we extend this construction to define H-minimal submanifolds in toric varieties.…”

Section: Hamiltonian-minimal Lagrangian Submanifolds In Toric Varietiesmentioning

confidence: 99%

Hamiltonian-minimal Lagrangian submanifolds in toric varieties

Mironov¹,

Panov²

2013

Russ. Math. Surv.

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“…where γ k1 > 0, c > 0, γ j2 > 0, γ i2 < 0 for 1 k m, 1 j p, p + 1 i m (this is the canonical form of an intersection of two quadrics described in [13,Proposition 4.2]). The second equation defines a cone over the product of two ellipsoids of dimensions 2p − 1 and 2q − 1.…”

Section: (42)mentioning

confidence: 99%

“…. , z i k are the only non-zero coordinates of z (see[13, Theorem 4.1]). Because in our case Z Γ contains points with only one non-zero coordinate, every γ i should generate the same lattice as the whole set γ 1 , .…”

mentioning

confidence: 99%

Minimal and H-minimal submanifolds in toric geometry

Kotelskiy

2016

Journal of Symplectic Geometry

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In this paper we investigate a family of Hamiltonian-minimal Lagrangian submanifolds in C m , CP m and other symplectic toric manifolds constructed from intersections of real quadrics. In particular we explain the nature of this phenomenon by proving H-minimality in a more conceptual way, and prove minimality of the same submanifolds in the corresponding moment-angle manifolds.1 by Y.Dong [5] and Hsiang-Lawson [7] we prove minimality of embeddings N ֒→ Z, and explain the underlying reasons for H-minimality for embeddings N ֒→ C m . Also, based on the note of A. Mironov and T.Panov [14] where they refer to the results of Y.Dong, we give an explicit construction and independent proof of H-minimality of Lagrangian submanifolds in symplectic toric manifolds, which generalize the result of A.Mironov.The paper is organized as follows: after introduction we give some background material on the notions of minimality and H-minimality. In the third section we review the construction of symplectic toric manifolds. The main results are stated in the last two sections. Minimality and H-minimality2.1. Minimality. Let M and L be smooth compact manifolds, g be a Riemannian metric on M . Assume that i : L ֒→ M is an embedding, i.e. L is a submanifold in M . We endow L with the metric induced from M .In this notation, we say that variation i t happens along the vector field X.Definition 2.2. An embedding i : L ֒→ M is called minimal if volume of L is stationary with respect to all variations, i.e.

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Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings

Cited by 23 publications

References 26 publications

Geometric structures on moment-angle manifolds

Geometric structures on moment-angle manifolds

Hamiltonian-minimal Lagrangian submanifolds in toric varieties

Minimal and H-minimal submanifolds in toric geometry

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