A bifurcation result for harmonic maps from an annulus to S^2 with not symmetric boundary data

We consider the problem of minimizing the energy of the maps u(r, theta) from the annulus Omega (rho) = B-1\(B) over bar (rho) to S-2 such that u(r, theta) is equal to (cos theta, sin theta, 0) for r = rho, and to (cos(theta + theta (0)), sin(theta + theta (0)), 0) for r = 1,where theta (0) is an element of [0, pi] is a fixed angle. We prove that the minimum is attained at a unique harmonic map u(rho) which is a planar map if log(2) rho + 3 theta (2)(0) less than or equal to pi (2), while it is not planar in the case log(2) rho + theta (2)(0) > pi (2). Moreover, we show that u(rho) tends to (v) over bar as rho --> 0, where (v) over bar minimizes the energy of the maps v(r, theta) from B-1 to S-2, with the boundary condition v(1, theta) = (cos(theta + theta (0)), sin(theta + theta (0)), 0).