Rigidity of isotropic maps (Q2365014)

The author studies the rigidity problem for isotropic harmonic maps from a compact Riemann surface to a complex projective space. The problem can be reduced to one in the algebraic category, involving certain curves of osculating spaces to a holomorphic curve. Theorem 3.8. Let \(f:X\longrightarrow {\mathbb{P}}^2\) be a holomorphic map from the Riemann surface \(X\) of genus \(g\geq 0\) so that \(f\) maps \(X\) birationally onto a curve \(Y=f(X)\) of degree \(d\) in the complex projective plane \({\mathbb{P}}^2\). Denote by \(r_1=\text{deg} (R_1)\) the total number of cusps of \(f\). If \(3d<2g-2-r_1\) then \(f\) is rigid. Proposition 4.13. If \(g(X)\geq 2\) and \(f:X\longrightarrow {\mathbb{P}}^2\) is an immersion of degree \(d\leq g-1\), then \(f\) is rigid. The author also presents a study of the ideal of associated curves and the curves \(f^{(k)}(X)\) (defined by the osculating \(k\)-planes to \(f\)).