On the Instability of a Minimal Surface in a 4-Manifold Whose Curvature Lies in the Interval (1/4, 1]

This theorem is related to the conjecture of Lawson and Simons [5], i.e., every minimal current in a complete simply connected Riemannian manifold whose curvature lies in the interval (1/4, 1] is unstable. The proof of this theorem can be roughly outlined as follows: First he shows that if a non-trivial cross-section u of the normal bundle v of M satisfies a certain differential equation ((*) in §1 in our terminology), then 6(u) + d(Ju) < 0, where J is the complex structure defined by the orientations of M and the ambient manifold. Secondly he constructs a solution of (*) on M which is homeomorphic to S. In this paper we investigate the dimension of the solution space H of (*) for a general immersed surface and obtain the following lemma.