Lagrangian Surfaces with Circle Symmetry in the Complex Two-Space

摘要

The hyper-Kaehler structure of the complex 2-space C~2 produces an interesting correspondence between minimal Lagrangian surfaces and holomorphic curves. Indeed, the correspondence is given by exchanging the orthogonal complex structure J to another one on the Euclidean 4-space R~4 [ChMo]. More generally, every Lagrangian conformal immersion of a Riemann surface M into C~2 can be transformed to a map from M (possibly from a covering of M) satisfying a Dirac-type equation with a specific potential term. It is an extension of the Cauchy-Riemann equation. This result is based on the following obvious facts. First, every immersion of M into C~2 is canonically identified with a section of the product vector bundle M x C~2 over M. Second, the "plus" part of the spin bundle S (associated to the canonical spin~C-structure induced from the complex structure on M) is isomorphic to the product complex line bundle M x C. In fact, the operator associated with the Dirac-type equation is the bona fide Dirac operator on the direct sum of the spin bundle S and its conjugate spin bundle S (see Section 1). Incidentally, it is now known that every conformal immersion of a Riemann surface into R~4 or R~3 can be expressed by a quaternionic analogue of the Cauchy-Riemann equation, and the quaternionic approach has advanced study for the surface theory (see [BFLPP]). However, we will study Lagrangian surfaces with prescribed Lagrangian angle in C~2, and description in terms of the complex numbers seems to be suitable for this study.