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

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.