The Space of Harmonic Maps from the 2-sphere to the Complex Projective Plane
نویسنده
چکیده
In this paper we study the topology of the space of harmonic maps from S to CP. We prove that the subspaces consisting of maps of a fixed degree and energy are path connected. By a result of Guest and Ohnita it follows that the same is true for the space of harmonic maps to CPn for n≥2. We show that the components of maps to CP are complex manifolds.
منابع مشابه
0 Se p 20 07 JACOBI FIELDS ALONG HARMONIC 2 - SPHERES IN S 3 AND S 4 ARE NOT ALL INTEGRABLE
In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4sphere which have non-integrable Jacobi fields. This is particularly surprising ...
متن کاملJacobi Fields along Harmonic 2 - Spheres in S 3 and S 4 Are Not All Integrable
In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4sphere which have non-integrable Jacobi fields. This is particularly surprising ...
متن کاملCR Singular Immersions of Complex Projective Spaces
Quadratically parametrized smooth maps from one complex projective space to another are constructed as projections of the Segre map of the complexification. A classification theorem relates equivalence classes of projections to congruence classes of matrix pencils. Maps from the 2-sphere to the complex projective plane, which generalize stereographic projection, and immersions of the complex pr...
متن کاملRegularity of the Space of Harmonic 2-spheres in the Unit 4-sphere
A harmonic map of the Riemann sphere into the unit 4-dimensional sphere has area 4πd for some positive integer d, and it is well-known that the space of such maps may be given the structure of a complex algebraic variety of dimension 2d+4. When d less than or equal to 2, the subspace consisting of those maps which are linearly full is empty. The twistor fibration from complex projective 3-space...
متن کاملM ar 1 99 6 On the space of harmonic 2 - spheres in C P 2
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 4πE is a smooth closed submanifold of the space of all C j maps (j ≥ 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 smooth and h...
متن کامل