Volume minimization for Lagrangian submanifolds in complex manifolds with negative first Chern class

2003/01/17 by Edward Goldstein, Goldstein, Edward
Mathematics · #53XX #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53XX

paper · pdf · doi:10.48550/arxiv.math/0301191

6 pages, minor modifications

arxiv created 2004/11/12 · arxiv updated 2009/11/30

Abstract

It is classically known that closed geodesics on a compact Riemann surface with a metric of negative curvature strictly minimize length in their free homotopy class. We'd like to generalize this to Lagrangian submanifolds in Kähler manifolds of negative Ricci curvature. The only known result in this direction is a theorem on Y.I. Lee for certain Lagrangian submanifolds in a product of two Riemann surfaces of constant negative curvature. We develop an approach to study this problem in higher dimensions. Along the way we prove some weak results (volume-minimization outside of a divisor) and give a counterexample to global volume-minimization for an immersed minimal Lagrangian submanifold.

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