Estimates on the stability of minimal surfaces and harmonic maps (Q1262558)

Let f: \(M\to N\) be a minimal immersion of a 2-dimensional simply- connected compact manifold M with piecewise \(C^ 1\) boundary \(\partial M\) into a Riemannian manifold N. Suppose that the sectional curvature of N is bounded and the sectional curvature of the Grassmann bundle \(G_ 2N\) is bounded from above. The author proves that there is a positive constant \(C_ 1\) depending only on N such that if the second fundamental form A of f satisfies \(\int_{M}(1+(1/2)| A|^ 2)dM < C_ 1,\) then M is stable. When f is a harmonic map, the stability condition of f is also obtained.