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Integral geometry and Hamiltonian volume minimizing property of a totally geodesic Lagrangian torus in \(S^ 2\times S^ 2\). (English) Zbl 1055.53063

The authors prove that the product of equators \(S^1\times S^1\) in \(S^2\times S^2\) is globally volume minimizing under Hamiltonian deformations. The proof is based on a Lagrangian intersection theorem for compact Hermitian symmetric spaces, and a new Poincaré formula for Lagrangian surfaces in \(S^2\times S^2\).
The result is in the spirit of a conjecture of Oh which reads as follows: if the fixed point set of an involutive anti-holomorphic isometry on a Kähler-Einstein manifold is a compact Einstein manifold with positive Ricci curvature, then it is globally volume minimizing under Hamiltonian deformations. The result of the paper under review shows that the statement of the conjecture is true for \(S^1\times S^1\subseteq S^2\times S^2\), even though \(S^1\times S^1\) is flat.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
53D12 Lagrangian submanifolds; Maslov index
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References:

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