The Bernstein problem for complete Lagrangian stationary surfaces

Mese, Chikako

doi:10.1090/S0002-9939-00-05603-3

The Bernstein problem for complete Lagrangian stationary surfaces
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Proc. Amer. Math. Soc. 129 (2001), 573-580 Request permission

Abstract:

In this paper, we investigate the global geometric behavior of lagrangian stationary surfaces which are lagrangian surfaces whose area is critical with respect to lagrangian variations. We find that if a complete oriented immersed lagrangian surface has quadratic area growth, one end and finite topological type, then it is minimal and hence holomorphic. The key to the proof is the mean curvature estimate of Schoen and Wolfson combined with the observation that a complete immersed surface of quadratic area growth, finite topology and $L^2$ mean curvature has finite total absolute curvature.

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