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

TL;DR

This paper proves that stable minimal surfaces conformally equivalent to the complex plane cannot exist in certain four-dimensional manifolds with positive isotropic curvature, extending to higher dimensions with positive complex sectional curvature.

Contribution

It establishes nonexistence results for stable minimal surfaces in manifolds with positive curvature conditions, generalizing previous theorems to higher dimensions.

Findings

No stable minimal surfaces conformally equivalent to the complex plane in 4-manifolds with positive isotropic curvature.

Extension of nonexistence results to higher-dimensional manifolds with positive complex sectional curvature.

Use of holomorphic bundle techniques and weighted L^2-methods in the proof.

Abstract

We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to the complex plane 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 Cauchy-Riemann operator of a holomorphic bundle, we prove (1) a vanishing theorem for these holomorphic bundles on the complex plane; (2) an existence theorem for holomorphic sections with controlled growth by Hormander's weighted L^2-method.