On closed surfaces with nonnegative curvature in the spectral sense

Abstract

We study closed orientable surfaces satisfying the spectral condition $\hat{I}{»}_1(-\hat{I}”+{\hat{I}}^{2}K)\ge \hat{I}»\ge 0$, where ${\hat{I}}^2$ is a positive constant and $K$ is the Gauss curvature. This condition naturally arises for stable minimal surfaces in 3-manifolds with positive scalar curvature. We show isoperimetric inequalities, area growth theorems and diameter bounds for such surfaces. The validity of these inequalities are subject to certain bounds for ${\hat{I}}^2$. Associated to a positive super-solution $\hat{I}”\ddot{I}†\le {\hat{I}}^{2}K\ddot{I}†$, the conformal metric $\ddot{I}{†}^{2/{\hat{I}}^2}g$ has pointwise nonnegative curvature. Utilizing the geometry of the new metric, we prove Hölder precompactness and almost rigidity results concerning the main spectral condition.

v2 updates: the article is largely re-written. Two main theorems are added. Section 2 is organized in a better way. Some original proofs in section 4 are simplified. The example in appendix B is replaced. Introduction and abstract are modified accordingly. The title is changed in order to avoid formulas. 26 pages. Comments are welcomed