Harmonic tori and generalised Jacobi varieties
Abstract
This article shows that every non-isotropic harmonic 2-torus in complex projective space factors through a generalised Jacobi variety related to the spectral curve. Each map is composed of a homomorphism into the variety and a rational map off it. The same ideas allow one to construct (pluri)-harmonic maps of finite type from Euclidean space into Grassmannians and the projective unitary groups. Further, some of these maps will be purely algebraic. For maps into complex projective space the algebraic maps of the plane are always doubly periodic i.e. they yield 2-tori. The classification of all these algebraic maps remains open.
Cite
@article{arxiv.math/9906076,
title = {Harmonic tori and generalised Jacobi varieties},
author = {Ian McIntosh},
journal= {arXiv preprint arXiv:math/9906076},
year = {2007}
}
Comments
LaTeX2e, 26 pages. Longer introduction for clarity, corrections throughout, slight re-arangement of material in section 3. To appear in: Communications in Analysis and Geometry