Rigidity of isotropic maps

Abstract

denote two isotropic harmonic maps from a compact Riemann surface to complex projective space. In this article we study whether from the isometry of / and g one may conclude their unitary equivalence. Using Calabi's rigidity theorem, this question may be reduced to one in the algebraic category, involving certain curves of osculating spaces to a holomorphic curve. We obtain some rigidity results mostly by analyzing the quadrics containing those curves. After recalling some definitions and basic facts, we show in §1 that our unitary question may be reduced to a projective one. Then in §2 we record some rigidity statements that follow easily from the use of projective invariants. In §3 and §4 we consider plane curves; we prove in (3.8) and (4.13) that ridigidy holds, roughly speaking, if the degree is small compared to the genus, providing a partial answer to a question posed by Quo-Shin Chi [C]. Motivated by the method of proof of Theorem (3.8), in §4 we begin a study of the ideal of associated curves and of the curves f^(X) introduced in §1. This is related to some aspects of Brill-Noether theory that we plan to pursue in another article. §1.