The L2 geometry of spaces of harmonic maps S2→S2 and RP2→RP2

✍ Scribed by J.M. Speight

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 243 KB
• Volume: 47
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(02)00227-9

✦ Synopsis

Harmonic maps from S 2 to S 2 are all weakly conformal, and so are represented by rational maps. This paper presents a study of the L 2 metric γ on M n , the space of degree n harmonic maps S 2 → S 2 , or equivalently, the space of rational maps of degree n. It is proved that γ is Kähler with respect to a certain natural complex structure on M n . The case n = 1 is considered in detail: explicit formulae for γ and its holomorphic sectional, Ricci and scalar curvatures are obtained, it is shown that the space has finite volume and diameter and codimension 2 boundary at infinity, and a certain class of Hamiltonian flows on M 1 is analysed. It is proved that Mn , the space of absolute degree n (an odd positive integer) harmonic maps RP 2 → RP 2 , is a totally geodesic Lagrangian submanifold of M n , and that for all n ≥ 3, Mn is geodesically incomplete. Possible generalizations and the relevance of these results to theoretical physics are briefly discussed.