NIRENBERG'S PROBLEM ON THE 2-DIMENSIONAL HEMI-SPHERE

In this paper, Morse theory under general boundary conditions is used to study the Nirenberg's problem on hemi-sphere : Given a smooth function K on hemi-sphere , with the standard metric, whether a conformal metric can be found, so that the Gaussian curvature equals to K and the boundary is again a geodesic curve.

It turns out that if deg (Ω, ∂_θK, 0) ≠ 1, where , ∂_nK is the exterior normal derivative and ∂_θK is the tangential derivative, then the Nirenberg's problem has a solution.