## Tohoku Mathematical Journal

### The space of harmonic two-spheres in the unit four-sphere

#### Abstract

A harmonic map of the Riemann sphere into the unit 4-dimensional sphere has area $4\pi\! d$ 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. We use the twistor fibration from complex projective 3-space to the 4-sphere to show that, if $d$ is equal to 3, 4 or 5, this subspace is a complex manifold.

#### Article information

Source
Tohoku Math. J. (2), Volume 58, Number 2 (2006), 231-236.

Dates
First available in Project Euclid: 22 August 2006

https://projecteuclid.org/euclid.tmj/1156256402

Digital Object Identifier
doi:10.2748/tmj/1156256402

Mathematical Reviews number (MathSciNet)
MR2248431

Zentralblatt MATH identifier
1106.58006

#### Citation

Bolton, John; Woodward, Lyndon M. The space of harmonic two-spheres in the unit four-sphere. Tohoku Math. J. (2) 58 (2006), no. 2, 231--236. doi:10.2748/tmj/1156256402. https://projecteuclid.org/euclid.tmj/1156256402

#### References

• J. L. Barbosa, On minimal immersions of $S^2$ into $S^2m$, Trans. Amer. Math. Soc. 210 (1975), 75--106.
• J. Bolton and L. M. Woodward, The space of harmonic maps of $S^2$ into $S^n$, Geometry and Global Analysis (Sendai, 1993), 165--173, Tohoku Univ., Sendai, 1993.
• J. Bolton and L. M. Woodward, Linearly full harmonic 2-spheres in $S^4$ of area $20 \pi$, Internat. J. Math. 12 (2001), 535--554.
• J. Bolton and L. M. Woodward, Higher singularities and the twistor fibration $\pi:\CC P^3\to S^4$, Geom. Dedicata 80 (2000), 231--245.
• R. L. Bryant, Conformal and minimal immersions of compact surfaces into the $4$-sphere, J. Differential Geom. 17 (1982), 455--473.
• S. S. Chern and J. Wolfson, Minimal surfaces by moving frames, Amer. J. Math. 105 (1983), 59--83.
• T. A. Crawford, The space of harmonic maps from the 2-sphere to the complex projective plane, Canad. Math. Bull. 40 (1997), 285--295.
• L. Fernandez, The dimension of the space of minimal 2-spheres in $S^6$, Preprint.
• M. Furuta, M. A. Guest, M. Kotani and Y. Ohnita, On the fundamental group of the space of harmonic 2-spheres in the $n$-sphere, Math. Z. 215 (1994), 503--518.
• H. B. Lawson, Jr., Surfaces minimales et la construction de Calabi-Penrose, Séminaire Bourbaki 1983/84, Astérisque 121--122 (1985), 197--211.
• B. Loo, The space of harmonic maps of $S^2$ into $S^4$, Trans. Amer. Math. Soc. 313 (1989), 81--102.
• L. Lemaire and J. C. Wood, On the space of harmonic 2-spheres in $\CC P^2$, Internat. J. Math. 7 (1996), 211--225.
• L. Lemaire and J. C. Wood, Jacobi fields along harmonic 2-spheres in $\CC P^2$ are integrable, J. London Math. Soc.(2) 66 (2002), 468--486.
• E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966.
• J.-L. Verdier, Two dimensional $\sigma$-models and harmonic maps from $S^2$ to $S^2n$, Lecture Notes in Phys. 180 (1982), 136--141.
• J.-L. Verdier, Applications harmoniques de $S^2$ dans $S^4$, Geometry Today (Rome, 1984), 267--282, Progr. Math. 60, Birkhäuser Boston, Boston, Mass., 1985.
• J.-L. Verdier, Applications harmoniques de $S^2$ dans $S^4$, II, Harmonic mappings, twistors, and $\sigma$-models (Luminy, 1986), 124--147, Adv. Ser. Math. Phys. 4, World Sci. Publishing, Singapore, 1988.