Minimal surfaces with constant curvature and Kaehler angle in complex space forms

Abstract

Minimal surfaces with constant Gaussian curvature in real space forms have been classifiedcompletely (cf. [Ca-2], [Ke-1], [Br] and their references). Next natural interest is to investigate minimal surfaces with constant Gaussian curvature in complex space forms, more generally in symmetric spaces. Prof. Kenmotsu posed the following problem: Classify minimal surfaces with constant Gaussian curvature in complex space forms. Recently, minimal 2-spheres with constant Gaussian curvature in complex projective spaces were classifiedindependently by [B-Oh] and [B-J-R-W]. [C-Z] studied pseudo-holomorphic curves of constant curvature in complex Grassmann manifolds. For an immersion <p of a Riemann surface M into a Kahler manifold N, the Kahler angle 0 of <pis defined to be the angle between Jd<p{d/dx) and d(p{d/dy), where z=x + V―ly is a local complex coordinate on M and / denotes the complex structure of N. Chern and Wolfson [Ch-W] pointed out the importance of the Kahler angle in the theory of minimal surfaces in Kahler manifolds. In [B-J-R-W] and [E-G-T] they investigated minimal 2-spheres in complex projective spaces and minimal surfaces in 2-dimensional complex space forms respectively in terms of the notion of Kahler angle. In this paper we classifyminimal surfaces with constant Gaussian curvature and constant Kahler angle in complex space forms.