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CLASSIFICATION OF LAGRANGIAN SURFACES OF CONSTANT CURVATURE IN THE COMPLEX EUCLIDEAN PLANE

Published online by Cambridge University Press:  23 May 2005

Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA (bychen@math.msu.edu)
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Abstract

One of the most fundamental problems in the study of Lagrangian submanifolds from a Riemannian geometric point of view is the classification of Lagrangian immersions of real-space forms into complex-space forms. In this article, we solve this problem for the most basic case; namely, we classify Lagrangian surfaces of constant curvature in the complex Euclidean plane $\mathbb{C}^2$. Our main result states that there exist 19 families of Lagrangian surfaces of constant curvature in $\mathbb{C}^2$. Twelve of the 19 families are obtained via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in $\mathbb{C}^2$ can be obtained locally from the 19 families.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2005

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CLASSIFICATION OF LAGRANGIAN SURFACES OF CONSTANT CURVATURE IN THE COMPLEX EUCLIDEAN PLANE
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