Boundary value problem of harmonic maps into ℂℙnand ℚℙn

Abstract

The boundary value problem for harmonic maps is solvable when the boundary image lies in a geodesic convex neighbourhood of the target manifold (see. [Ha], [H-K-W]). In general, a size restriction of the boundary map is necessary and the result in [H-K-W] is optimal. But, one still expects the solvability for the boundary value problem with large image range when the boundary condition is ‘sufficiently nice’. In [J-K] and [E-L1] the authors consider the rotationally symmetric harmonic maps from B into S whose boundary values lie just outside of a geodesic convex neighbourhood. Recently, many works have been written on maps from B into S (see [Ha], [H-K-L1], [H-K-L2], [H-L-P] and [Z]). It is natural to investigate the problem with large boundary data when the target manifold with variable sectional curvature. The first candidate is complex projective space with the Fubini-Study metric. In author’s previous work [X2] a reduction method for large and equivariant boundary data has been exhibited, and the simplist case has been solved by the heat flow method. In the present paper we study more general situations by the variational method. Besides the concrete results, the research on this problem might give some implications on the obstruction of the solvability of certain boundary value problems for a system of elliptic PDE. Let B be the (m+1)-dimensional unit ball. Under an Sm−1 action the base region D ∈ R is given by