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Transactions of the American Mathematical Society

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On minimal surfaces in a Kähler manifold of constant holomorphic sectional curvature

Author: Jon G. Wolfson
Journal: Trans. Amer. Math. Soc. 290 (1985), 627-646
MSC: Primary 53C42; Secondary 58E20
MathSciNet review: 792816
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Abstract: This paper studies minimal surfaces in Kähler manifolds of constant holomorphic sectional curvature using the technique of the moving frame. In particular, we provide a classification of the minimal two-spheres in $ {\mathbf{C}}{P^n}$, complex projective $ n$-space, equipped with the Fubini-Study metric. This classification can be described as follows: To each holomorphic curve in $ {\mathbf{C}}{P^n}$ classically there is associated a particular framing of $ {{\mathbf{C}}^{n + 1}}$ called the Frenet frame. Each element of the Frenet frame induces a minimal surface in $ {\mathbf{C}}{P^n}$. The classification theorem states that all minimal surfaces of topological type of the two-sphere occur in this manner. The theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in Kähler manifolds of constant holomorphic sectional curvature together with the Riemann-Roch Theorem.

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Article copyright: © Copyright 1985 American Mathematical Society

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