On complete stable minimal surfaces in 4-manifolds with positive isotropic curvature

Abstract:

We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to $\mathbb {C}$ in any complete orientable four- dimensional Riemannian manifold with uniformly positive isotropic curvature. We also generalize the same nonexistence result to higher dimensions provided that the ambient manifold has uniformly positive complex sectional curvature. The proof consists of two parts: assuming an “eigenvalue condition” on the $\overline {\partial }$-operator of a holomorphic bundle, we prove (1) a vanishing theorem for these holomorphic bundles on $\mathbb {C}$ and (2) an existence theorem for holomorphic sections with controlled growth by Hörmander’s weighted $L^2$-method.