Some estimates related to Oh's conjecture for the Clifford tori in CPⁿ

2003/11/26 by Edward Goldstein, Goldstein, Edward
Mathematics · #53XX #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:53XX

paper · pdf · doi:10.48550/arxiv.math/0311460

arxiv created 2003/11/26 · openalex publication_date 2003/11/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This note is motivated by Y.G. Oh's conjecture that the Clifford torus L_n in ℂPⁿ minimizes volume in its Hamiltonian deformation class. We show that there exist explicit positive constants a_n depending on the dimension with a₂=3/π such that for any Lagrangian torus L in the Hamiltonian class of L_n we have vol(L) ≥ a_n vol (L_n). The proof uses the recent work of C.H. Cho on Floer homology of the Clifford tori. A formula from integral geometry enables us to derive the estimate. We wish to point out that a general lower bound on the volume of L exists from the work of C. Viterbo. Our lower bound a₂= 3/π is the best one we know.

Citations

• The kinematic formula in Riemannian homogeneous spaces

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