On the space of harmonic $2$-spheres in ${\bf C}P^2$

Details On the space of harmonic $2$-spheres in ${\bf C}P^2$ On the space of harmonic $2$-spheres in ${\bf C}P^2$

1995-10-06

arxiv.org/abs/dg-ga/9510003v2

Mathematics

Differential Geometry

Minor revision to take into account improved result of T.A. Crawford. Abstract and p. 2 changed plus some typos corrected. 15

Scientific paper

Carrying further work of T.A. Crawford, we show that each component of the space of harmonic maps from the $2$-sphere to complex projective $2$-space of degree $d$ and energy $4 \pi E$ is a smooth closed submanifold of the space of all $C^j$ maps $(j \geq 2)$. We achieve this by showing that the Gauss transform which relates them to spaces of holomorphic maps of given degree and ramification index is {\bf smooth} and has {\bf injective differential}.

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