@MISC{Panov12intersectionsof,

author = {Taras Panov},

title = {Intersections of quadrics and H-minimal Lagrangian submanifolds},

year = {2012}

}

An immersion i: N � M of an n-manifold N is Lagrangian if i ∗ (ω) = 0. If i is an embedding, then i(N) is a Lagrangian submanifold of M. A vector field ξ on M is Hamiltonian if the 1-form ω ( · , ξ) is exact. A Lagrangian immersion i: N � M is Hamiltonian minimal (H-minimal) if the variations of the volume of i(N) along all Hamiltonian vector fields with compact support are zero, i.e. d dt vol(it(N)) ∣ ∣ t=0 = 0, where it(N) is a Hamiltonian deformation of i(N) = i0(N). Taras Panov (MSU) Quadrics and Lagrangian submanifolds Zlatibor 7 Sep 2012 2 / 19Overview Explicit examples of H-minimal Lagrangian submanifolds in C m and CP m were constructed in the work of Yong-Geun Oh, Castro–Urbano, Hélein–Romon, Amarzaya–Ohnita, among others. In 2003 Mironov suggested a universal construction providing an H-minimal Lagrangian immersion in C m from an intersection of special real quadrics. The same intersections of real quadrics are known to toric geometers and topologists as (real) moment-angle manifolds. They appear, for instance, as the level sets of the moment map in the construction of Hamiltonian toric manifolds via symplectic reduction. Here we combine Mironov’s construction with the methods of toric topology to produce new examples of H-minimal Lagrangian embeddings with interesting and complicated topology. Taras Panov (MSU) Quadrics and Lagrangian submanifolds Zlatibor 7 Sep 2012 3 / 19Polytopes and moment-angle manifolds A convex polytope in R n is obtained by intersecting m halfspaces: P = { x ∈ R n: 〈ai, x 〉 + bi � 0 for i = 1,..., m}. Suppose each Fi = P ∩ {x: 〈ai, x 〉 + bi = 0} is a facet (m facets in total). Define an affine map iP: R n → R m, iP(x) = () 〈a1, x 〉 + b1,..., 〈am, x 〉 + bm. Then iP is monomorphic, and iP(P) ⊂ R m is the intersection of an n-plane with R m � = {y = (y1,..., ym): yi � 0}.

h-minimal lagrangian submanifolds lagrangian submanifolds zlatibor moment-angle manifold universal construction lein romon convex polytope vector field h-minimal lagrangian immersion affine map ip hamiltonian toric manifold mironov construction hamiltonian vector field moment map castro urbano real quadric tara panov lagrangian submanifold compact support amarzaya ohnita lagrangian immersion hamiltonian deformation new example toric topology yong-geun oh special real quadric dt vol h-minimal lagrangian embeddings complicated topology level set symplectic reduction explicit example

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