Holomorphic Embedding of Complex Curves in Spaces of Constant Holomorphic Curvature - Chavel, Issac (1972)
A special case of Wirtinger's theorem asserts that a complex curve (two-dimensional) holomorphically embedded in a Kaehler manifold is a minimal surface. The converse is not necessarily true. Guided by considerations from the theory of moduli of Riemann surfaces, we discover (among other results) sufficient topological and differential-geometric conditions for a minimal (Riemannian) immersion of a 2-manifold in complex projective space with the Fubini-Study metric to be holomorphic.